structures:matrix:discussion
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| We can write them like this: \(\ \left[\matrix{1 & 4 & x \cr 0 & \frac12 & 2}\right],\ \) which is a \(\ 2{\times}3\ \) matrix. | We can write them like this: \(\ \left[\matrix{1 & 4 & x \cr 0 & \frac12 & 2}\right],\ \) which is a \(\ 2{\times}3\ \) matrix. | ||
| - | Most data sets we look at through to statistics in year 12 can be thought of as a //Matrix//. <wrap #vectormatrix/>A //Vector// can also be thought of as a single row or single column //Matrix//. Each //Real Function// can be thought of as an infinite-dimensional vector, but that is way beyond high school. Matrices are much more powerful than that! Square matrices can represent and perform the transformations we look at in //Cartesian Space// and in functions --- //Linear// ones like the rotations, reflections, stretches and skews which preserve `0` and the more general //Affine// ones like translations. And the Complex Numbers (including the Reals) can be thought of as particular 2\(\times\)2 matrices --- which match the linear transformations in 2D space in the very intriguing way I will [[#complex-as-matrix|work towards]] in this discussion. But there is much to fill in first. | + | Most data sets we look at through to statistics in year 12 can be thought of as a //Matrix//. <wrap #vectormatrix/>A //Vector// can also be thought of as a single row or single column //Matrix//. Each //Real Function// [[#functions|can be thought of as an infinite-dimensional vector]], but that is way beyond high school. Matrices are much more powerful than that! Square matrices can represent and perform the transformations we look at in //Cartesian Space// and in functions --- //Linear// ones like the rotations, reflections, stretches and skews which preserve `0` and the more general //Affine// ones like translations. And the Complex Numbers (including the Reals) can be thought of as particular 2\(\times\)2 matrices --- which match the linear transformations in 2D space in the very intriguing way I will [[#complex-as-matrix|work towards]] in this discussion. But there is much to fill in first. |
| ====Scalar==== | ====Scalar==== | ||
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| Can you frame an equation or two that answers it more directly. This is about understanding the systems we are describing and how the mathematical language we use in that description works. There are plenty of maths packages where you could type in the equation we might make to model the question and the answer would just be be returned. We describe systems in this way and being able to read the equation and 'see' the system is fluency with this language. Problems are often not so neat and more purely vector solutions can be needed. It is the component of the force along the track that is going to accelerate the train. | Can you frame an equation or two that answers it more directly. This is about understanding the systems we are describing and how the mathematical language we use in that description works. There are plenty of maths packages where you could type in the equation we might make to model the question and the answer would just be be returned. We describe systems in this way and being able to read the equation and 'see' the system is fluency with this language. Problems are often not so neat and more purely vector solutions can be needed. It is the component of the force along the track that is going to accelerate the train. | ||
| - | A deep understanding of the idea of dot product --- its different meanings and methods, this equivalence and its relationship to spaces, functions and equations is needed for several extension maths problem types that always turn up. It has a wide range of meanings beyond what we see in school, including the Matrix ideas behind this discussion. | + | A deep understanding of the idea of dot product --- its different meanings and methods, this equivalence and its relationship to spaces, [[#functions]] and equations is needed for several extension maths problem types that always turn up. It has a wide range of meanings beyond what we see in school, including the Matrix ideas behind this discussion. |
| ====Space==== | ====Space==== | ||
| - | This idea we have also developed in high school. In geometry we think about lines, points, directions, angles, distances and surfaces. We think (mostly) about 2 or 3 dimensions and areas, volumes and such. Then we think about **Cartesian Space**, describing positions using axes as we did [[#cartesianvector|above]]. We get into describing shapes with functions and finding directions, slopes, areas and volumes with geometrical techniques and with calculus. We start to think about how vectors fit in with these ideas. We also think about transformations of shapes in this space: rotations, reflections, stretching in one or all directions and sliding in any direction. There is a [[linear#linear transformations with matrices|great deal more]] to this idea of transformation than we have met so far!! | + | This idea we have also developed in high school. In geometry we think about lines, points, directions, angles, distances and surfaces. We think (mostly) about 2 or 3 dimensions and areas, volumes and such. Then we think about **Cartesian Space**, describing positions using axes as we did [[#cartesianvector|above]]. We get into describing shapes with **[[#Functions]]** and finding directions, slopes, areas and volumes with geometrical techniques and with calculus. We start to think about how vectors fit in with these ideas. We also think about transformations of shapes in this space: rotations, reflections, stretching in one or all directions and sliding in any direction. There is a [[linear#linear transformations with matrices|great deal more]] to this idea of transformation than we have met so far!! |
| An important idea is **Euclidean Space**, which essentially means an ideal of //flatness// in the sense that //parallel lines// and our common idea of //distance between two points// make the usual geometrical sense. In Euclidean Space our familiar geometry applies, especially ideas like Pythagoras Rule about sides of right angle triangles and the angles of a triangle adding up to a straight angle. In reality our universe is not so simple, the space we exist in is not flat. Classical geometry is only an approximation. A very good one if size and temperature are in the range familiar to our senses. Time is not what it seems and the whole thing is springy so that we get 'gravity waves' travelling through the fabric of space itself. When things get big enough, small enough, energetic enough or cold enough we need to drastically rework our warm and human-scale intuition. Matrices are an important way to enable that. | An important idea is **Euclidean Space**, which essentially means an ideal of //flatness// in the sense that //parallel lines// and our common idea of //distance between two points// make the usual geometrical sense. In Euclidean Space our familiar geometry applies, especially ideas like Pythagoras Rule about sides of right angle triangles and the angles of a triangle adding up to a straight angle. In reality our universe is not so simple, the space we exist in is not flat. Classical geometry is only an approximation. A very good one if size and temperature are in the range familiar to our senses. Time is not what it seems and the whole thing is springy so that we get 'gravity waves' travelling through the fabric of space itself. When things get big enough, small enough, energetic enough or cold enough we need to drastically rework our warm and human-scale intuition. Matrices are an important way to enable that. | ||
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| We do not look much at non-euclidean spaces in school --- the main one is looking at the surface of the earth, and in particular different maps of the world, different projections of the globe onto a flat piece of paper. Scales and straight lines get messed up, there needs to be stitching together, nearby places can be on the other side of cuts and a long way apart on the paper. Parallel lines do not make sense to a pilot of a jetliner. Pythagoras rule fails on triangles, the angle sum rule fails. We have a very different geometry. And then we see it embedded in 3-space, causing day and night, winter and summer, midnight sun in the arctic summer, timezones and the dateline. | We do not look much at non-euclidean spaces in school --- the main one is looking at the surface of the earth, and in particular different maps of the world, different projections of the globe onto a flat piece of paper. Scales and straight lines get messed up, there needs to be stitching together, nearby places can be on the other side of cuts and a long way apart on the paper. Parallel lines do not make sense to a pilot of a jetliner. Pythagoras rule fails on triangles, the angle sum rule fails. We have a very different geometry. And then we see it embedded in 3-space, causing day and night, winter and summer, midnight sun in the arctic summer, timezones and the dateline. | ||
| - | Understanding spaces which are not flat we make a great deal of use of the idea of a //Smooth// space --- where when we zoom in close enough it looks very much like Euclidean Space. There are no kinks, gaps, sudden jumps or hard corners. This is greatly helped by the ideas of **Infinitesimal Calculus** that we get a start on in high school. We can think of the typical functions we look at in calculus as describing a one dimensional space that mostly is not flat. Drawing a graph could be considered visualising embedding this space in the two dimensional Cartesian Plane. One way we developed calculus was zooming in closer and closer using the mathematical ideas of //Limits// and small changes --- deltas like `Delta t` becoming infinitesimals like \({\rm d}t\). We found straight line or rectangle limits to describe the slopes or areas related to a curvy space described by a function and made new functions that gives a meaningful and useful answers to the questions "What is the slope of this space?", "What is the curvature of this space?" and a way of going backwards on this that turns out to answer very different question about changes or cumulative changes in the position of this space as embedded in the larger space. Also questions like "How much area is there under a part of this space, in the higher dimensional euclidean space we embedded it in?". Another way to say these are "How fast are things changing at a particular time?" (Derivative of a function if time is the function input) and "What is the total change during an interval of time?" (Definite Integral, of a function describing rate of change with time as its input). These ideas were first developed to describe movement and forces in Euclidean Space with an eternally ticking universal background time. They proved incredibly useful in many kinds of analysis about things that change. This completely new understanding of the world enabled the industrial revolution, and much of what we think of as a comfortable life. | + | Understanding spaces which are not flat we make a great deal of use of the idea of a //Smooth// space --- where when we zoom in close enough it looks very much like Euclidean Space. There are no kinks, gaps, sudden jumps or hard corners. This is greatly helped by the ideas of **Infinitesimal Calculus** that we get a start on in high school. We can think of the typical functions we look at in calculus as describing a one dimensional space that mostly is not flat. Drawing a [[#graph of a function]] could be considered visualising embedding this space in the two dimensional Cartesian Plane. One way we developed calculus was zooming in closer and closer using the mathematical ideas of //Limits// and small changes --- deltas like `Delta t` becoming infinitesimals like \({\rm d}t\). We found straight line or rectangle limits to describe the slopes or areas related to a curvy space described by a function and made new functions that gives a meaningful and useful answers to the questions "What is the slope of this space?", "What is the curvature of this space?" and a way of going backwards on this that turns out to answer very different question about changes or cumulative changes in the position of this space as embedded in the larger space. Also questions like "How much area is there under a part of this space, in the higher dimensional euclidean space we embedded it in?". Another way to say these are "How fast are things changing at a particular time?" (Derivative of a function if time is the function input) and "What is the total change during an interval of time?" (Definite Integral, of a function describing rate of change with time as its input). These ideas were first developed to describe movement and forces in Euclidean Space with an eternally ticking universal background time. They proved incredibly useful in many kinds of analysis about things that change. This completely new understanding of the world enabled the industrial revolution, and much of what we think of as a comfortable life. |
| ===Vector Space=== | ===Vector Space=== | ||
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| That is just the start. We can consider polynomials as vectors (the coefficients are the list of scalars). <wrap #function-as-vector/>We can even consider **[[#functions]]** that give a scalar for each element in some domain as vectors, we can add them and we can multiply them by a scalar and it works the same way as it does with vectors --- thinking of the dimension of this one, it is big! Each such function an infinite dimensional vector. And functions do behave like vectors in this way. It can be a very useful for generalising and understanding functions as abstract entities. | That is just the start. We can consider polynomials as vectors (the coefficients are the list of scalars). <wrap #function-as-vector/>We can even consider **[[#functions]]** that give a scalar for each element in some domain as vectors, we can add them and we can multiply them by a scalar and it works the same way as it does with vectors --- thinking of the dimension of this one, it is big! Each such function an infinite dimensional vector. And functions do behave like vectors in this way. It can be a very useful for generalising and understanding functions as abstract entities. | ||
| - | In school, Extension 2 only, we meet the //Cross Product//. It is a 3D-only special case of the //Exterior Product//. The magnitude of the //Cross Product// is the area of the parallelogram formed by the two vectors, \(\ |{\bf v\times w}|=|{\bf v}| |{\bf w}|\sin\theta.\ \) It is complementary to Dot Product when considered geometrically, it uses sine rather than cosine. Dot Product is zero when perpendicular while Cross Product is zero when parallel. But Cross Product is a vector, its direction is perpendicular to each operand and up or down according to the Right Hand Rule. It is an area and a direction, an oriented area. The product of two directed lengths making a directed area fits with our geometrical ideas of area. The polar form comes directly from that. This Right Hand Rule business means it is not cleanly a well behaved vector. Sometimes we call it a //Pseudovector//, more generally we speak of a //Bivector//. Equivalently, working in cartesian form we find that | + | We no longer look at the //Cross Product// in school. It is a 3D-only special case of the //Exterior Product//. The magnitude of the //Cross Product// is the area of the parallelogram formed by the two vectors, \(\ |{\bf v\times w}|=|{\bf v}| |{\bf w}|\sin\theta.\ \) It is complementary to Dot Product when considered geometrically, it uses sine rather than cosine. Dot Product is zero when perpendicular while Cross Product is zero when parallel. But Cross Product is a vector, its direction is perpendicular to each operand and up or down according to the Right Hand Rule. It is an area and a direction, an oriented area. The product of two directed lengths making a directed area fits with our geometrical ideas of area. The polar form comes directly from that. This Right Hand Rule business means it is not cleanly a well behaved vector. Sometimes we call it a //Pseudovector//, more generally we speak of a //Bivector//. Equivalently, working in cartesian form we find that |
| \[\ \left[\matrix{v_1\cr v_2\cr v_3}\right]\times\left[\matrix{w_1\cr w_2\cr w_3}\right]= | \[\ \left[\matrix{v_1\cr v_2\cr v_3}\right]\times\left[\matrix{w_1\cr w_2\cr w_3}\right]= | ||
| \left[\matrix{v_2 w_3 - v_3 w_2\cr v_1 w_3 - v_3 w_1\cr v_1 w_2 - v_2 w_1}\right]\] | \left[\matrix{v_2 w_3 - v_3 w_2\cr v_1 w_3 - v_3 w_1\cr v_1 w_2 - v_2 w_1}\right]\] | ||
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| \[({\bf a\,b\,c})={\bf a}\bullet({\bf b\times c})=\left|\matrix{a_1&a_2&a_3\cr b_1&b_2&b_3\cr c_1&c_2&c_3}\right|\] | \[({\bf a\,b\,c})={\bf a}\bullet({\bf b\times c})=\left|\matrix{a_1&a_2&a_3\cr b_1&b_2&b_3\cr c_1&c_2&c_3}\right|\] | ||
| - | This suggests extending to more dimensions --- well beyond the scope of our school of course. The //Determinant// has several interpretations. The idea is millenia old, imported into europe along with algebra a bit over 500 years ago. The european name is well before Matrices, they were developed to support linear algebra. //Determinant// comes from linear algebra, in part studying the simultaneous equations we touch on early in our school algebra. It appears in the "The Nine Chapters on the Mathematical Art" the classical chinese mathematical text, a compilation of older knowledge in their typical classical style, published around 2300 years ago. It shows whether the equations have a unique solution. The determinant is also closely related to eigenvalues and eigenvectors and in turn deeply part of quantum mechanics where Complex Matrices play a large role. It is often more important as a theoretical rather than practical method --- there are more efficient approaches to most actual calculations. It is about understanding a system in abstract, and matrices are fundamental to the description of the cosmos and the descriptions on atomic scale. And to modelling any big system, say climate or business logistics. And to computation, hence most of the technology around us. | + | This suggests extending to more dimensions. The //Determinant// has several interpretations. The idea is millenia old, imported into europe along with algebra a bit over 500 years ago. The european name is well before Matrices, they were developed to support linear algebra. //Determinant// comes from linear algebra, in part studying the simultaneous equations we touch on early in our school algebra. It appears in the "The Nine Chapters on the Mathematical Art" the classical chinese mathematical text, a compilation of older knowledge in their typical classical style, published around 2300 years ago. It shows whether the equations have a unique solution. The determinant is also closely related to eigenvalues and eigenvectors and in turn deeply part of quantum mechanics where Complex Matrices play a large role. It is often more important as a theoretical rather than practical method --- there are more efficient approaches to most actual calculations. It is about understanding a system in abstract, and matrices are fundamental to the description of the cosmos and the descriptions on atomic scale. And to modelling any big system, say climate or business logistics. And to computation, hence most of the technology around us. |
| <WRAP notify> | <WRAP notify> | ||
| We are starting to see here how **[[#Space]], [[#Scalar]], [[#Vectors]]** and **[[#Matrices]]** fit together. We have a vector, \(\bf v\), in `n`-space. We can describe it as an arrow, a magnitude with a direction. We can describe the same entity as a list of scalars, `v_i`. In another interpretation it is a single column or single row matrix `V` and can participate in matrix operations. Each of these representations can also describe a position in `n`-dimensional geometrical space. | We are starting to see here how **[[#Space]], [[#Scalar]], [[#Vectors]]** and **[[#Matrices]]** fit together. We have a vector, \(\bf v\), in `n`-space. We can describe it as an arrow, a magnitude with a direction. We can describe the same entity as a list of scalars, `v_i`. In another interpretation it is a single column or single row matrix `V` and can participate in matrix operations. Each of these representations can also describe a position in `n`-dimensional geometrical space. | ||
| - | Position in one dimensional space can be described by a scalar, often a Real Number. Direction here is //sign//, positive or negative, forward or back. That is our very familiar number line --- we need to pick a 'zero' point and also a 'one' point which gives the scalar its 'unit'. Then we can use numbers to describe position. Complex Numbers, another kind of scalar, do not fit on a line. They do not have that kind of 'order'. But they do behave like numbers in other ways, \({+}{\times}{=}{1}{\div}{0}{-}\) all behave properly. \(\ b^a\sin a\,\log_b a\ \) and number algebra are better behaved than Real Numbers. They have this extra number, \(\ \rm i.\,\) We can place it distance `1` from 'zero' perpendicular to the number line. We pick that point \(\,\rm i\,\) and we see the Complex Numbers as all the points on a Number Plane. Now we have a magnitude and direction sense of these numbers. Direction here is angle rather than sign. It no longer defines an 'order', it is a different kind of positional relationship instead. Numbers `pi` and \(\rm e\) emerge as fundamental to this system, linking its geometrical and component representations. The trig and exponential functions are deeply involved here. | + | Position in one dimensional space can be described by a scalar, often a Real Number. Direction here is //sign//, positive or negative, forward or back. That is our very familiar number line --- we need to pick a 'zero' point and also a 'one' point which gives the scalar its 'unit'. Then we can use numbers to describe position. Complex Numbers, another kind of scalar, do not fit on a line. They do not have that kind of 'order'. But they do behave like numbers in other ways, \({+}{\times}{=}{1}{\div}{0}{-}\) all behave properly. \(\ b^a\sin a\,\log_b a\ \) and number algebra are better behaved than Real Numbers. They have this extra number, \(\ \rm i.\,\) We can place it distance `1` from 'zero' perpendicular to the number line. We pick that point \(\,\rm i\,\) and we see the Complex Numbers as all the points on a Number Plane. Now we have a magnitude and direction sense of these numbers. Direction here is angle rather than sign. It no longer defines an 'order', it is a different kind of positional relationship instead. Numbers `pi` and \(\rm e\) emerge as fundamental to this system, linking its geometrical and component representations. The trig and exponential [[#functions]] are deeply involved here. |
| A scalar which we can also see as 2D, things are getting interesting. Vectors or Cartesian points can represent a plane, but \(\ {\times}{\div}a^b\sin\log\ \)do not make much sense with vectors or with geometrical points. They are perfectly sensible as Complex Number operations. We are pulling together algebra, geometry and trigonometry, creating a single framework. It turns out that nature is quite happy with this kind of quantity, several values we can measure make a great deal of sense as Complex Numbers. But that does challenge our geometrical intuition, and we need to develop our mathematical language to support this. And to support the fact that space and time are not simple, flat and independent in the way that geometrical intuition derived from our senses and human-scale experience might suggest. Here matrices become very important. | A scalar which we can also see as 2D, things are getting interesting. Vectors or Cartesian points can represent a plane, but \(\ {\times}{\div}a^b\sin\log\ \)do not make much sense with vectors or with geometrical points. They are perfectly sensible as Complex Number operations. We are pulling together algebra, geometry and trigonometry, creating a single framework. It turns out that nature is quite happy with this kind of quantity, several values we can measure make a great deal of sense as Complex Numbers. But that does challenge our geometrical intuition, and we need to develop our mathematical language to support this. And to support the fact that space and time are not simple, flat and independent in the way that geometrical intuition derived from our senses and human-scale experience might suggest. Here matrices become very important. | ||
| </WRAP> | </WRAP> | ||
| - | Consider a scalar, \(s,\) as the one-element matrix \(S=\left[s\right]\) and as a 1D vector \(\bf s\) with magnitude `|s|`, as above. Consider \(\bf v\) and corresponding single column matrix `V,` \(s{\bf v}\) is \(SV.\) Consider another vector `W`. `W^T` is then a row vector, \({\bf w\bullet v}\) is \(W^TV\) if the single entry matrix result is considered a scalar. Look [[applications#working with matrices|here]] for more detail on these matrix operations --- I have details a little outside this page since here I am interested in exploring and extending the ideas covered in the HSC. | + | Consider a scalar, \(s,\) as the one-element matrix \(S=\left[s\right]\) and as a 1D vector \(\bf s\) with magnitude `|s|`, as above. Consider \(\bf v\) and corresponding single column matrix `V,` \(s{\bf v}\) is \(SV.\) Consider another vector `W`. `W^T` is then a row vector, \({\bf w\bullet v}\) is \(W^TV\) if the single entry matrix result is considered a scalar. Look [[applications#working with matrices|here]] for more detail on these matrix operations --- I have details outside this page since here I am mostly interested in exploring and extending the ideas covered in the HSC. |
| If convenient we can just work with Matrices, scalars are a single entry matrix, vectors are a single column matrix, their transpose is a single row matrix, scalar multiplication and dot product are defined as above. There is much more that can be done to extend these with matrices. | If convenient we can just work with Matrices, scalars are a single entry matrix, vectors are a single column matrix, their transpose is a single row matrix, scalar multiplication and dot product are defined as above. There is much more that can be done to extend these with matrices. | ||
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| ====Fields==== | ====Fields==== | ||
| - | This idea we touch on in high school, the idea that every point in a space might be associated with a value. There is another very important mathematical meaning for the word //Field//, quite different, referring to the kind of numbers which allow division for any pair except dividing by `0` --- which are a very important kind of scalar, well known examples are Rational, Real and Complex Numbers --- the scalars of a Vector Space must be this kind but I avoided using the word because this new usage was coming up. Initially these values are scalars in a Scalar Field, generally our familiar real numbers. We do not use the word //field// at first, but graphing a function in a 2D space is really showing a //Scalar Field// on a one dimensional space by adding another axis to show the value at each point --- as distance above the number line. It is a very powerful way of imagining and working with a function. A function is a description of a field if the input can be considered the coordinates in a space, since then it assigns a unique value to each point. A lot of the last three years at school deals with this! | + | This idea we touch on in high school, the idea that every point in a space might be associated with a value. There is another very important mathematical meaning for the word //Field//, quite different, referring to the kind of numbers which allow division for any pair except dividing by `0` --- which are a very important kind of scalar, well known examples are Rational, Real and Complex Numbers --- the scalars of a Vector Space must be this kind but I avoided using the word because this new usage was coming up. Initially these values are scalars in a Scalar Field, generally our familiar real numbers. We do not use the word //field// at first, but a [[#graph of a function]] in a 2D space is really showing a //Scalar Field// on a one dimensional space by adding another axis to show the value at each point --- as distance above the number line. It is a very powerful way of imagining and working with a function. A function is a description of a field if the input can be considered the coordinates in a space, since then it assigns a unique value to each point. A lot of the last three years at school deals with this! |
| But we are also familiar with a **Vector Field**, and certainly deal with them in high school --- the wind map that the weather bureau puts out is a Vector Field on two dimensions, every point on the map has a wind speed and direction so every point has a vector associated with it and we draw that with little arrows for the vectors placed on their points in the map. This is a very important kind of analysis used in engineering, manufacturing, computation, planning, predicting complex behaviors and understanding the results of studies. A lot of the core university maths is the calculus that deals with that, looking at rates of change of those vectors and sums of those vectors in different ways. An extension of the way school calculus looks at rates of change and sums of velocity in projectile motion. | But we are also familiar with a **Vector Field**, and certainly deal with them in high school --- the wind map that the weather bureau puts out is a Vector Field on two dimensions, every point on the map has a wind speed and direction so every point has a vector associated with it and we draw that with little arrows for the vectors placed on their points in the map. This is a very important kind of analysis used in engineering, manufacturing, computation, planning, predicting complex behaviors and understanding the results of studies. A lot of the core university maths is the calculus that deals with that, looking at rates of change of those vectors and sums of those vectors in different ways. An extension of the way school calculus looks at rates of change and sums of velocity in projectile motion. | ||
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| ====Functions==== | ====Functions==== | ||
| - | Consider our familiar functions like \({\rm f}(x)\) giving a real number for each real number. We can view this collection as a Vector Space which includes all possible examples of such a function. It has infinite dimension, and not just the infinite of the integers. In that case each would be an endless list of values one after the other. But this is the much bigger infinite of the number of real numbers. These vectors have an independent component for every distinct input. Each different list of such components describes a different function. The hypothetical list defines each output given the input. It is a way to give the function mapping rule. A very long and dense list, thats for sure! But that is an infinitely long, infinitely dense list of real numbers otherwise nicely ordered (like Reals are). As a vector there is a distinct axis or direction for each potential distinct input, whatever that might mean. | + | Consider our familiar functions like \({\rm f}(x)\) giving a real number for each real number. We can view this collection as a Vector Space which includes all possible examples of such a function. It has infinite dimension. Not just the infinite of the integers, in that case each would be an endless list of values one after the other. But this is the much bigger infinite of the number of real numbers. These vectors have an independent component for every distinct input. Each different list of such components describes a different function. The hypothetical list defines each output given the input. It is a way to give the function mapping rule. A very long and dense list, thats for sure! But that is an infinitely long, infinitely dense list of real numbers otherwise nicely ordered (like Reals are). As a vector there is a distinct axis or direction for each potential distinct input, whatever that might mean. |
| - | ===graph of function=== | + | ===graph of a function=== |
| - | Imagine a graph, the number line with graph above or below it as given by the output of one of the functions in that collection. The usual kind of graph we draw. Now consider that graph as a list of values along the number line. The value 'listed' for each point on the horizontal axis is the height above that axis of the line we drew. Some nice, detailed list of values indeed! With a value given for each point on the number line the dimension is the size of the set of all real numbers. [[#fields|Earlier]] we had thought of such a function as a Scalar [[#Fields|Field]], or thought of a Scalar Field as a function. We can alternatively consider this curved line as a curved one dimensional [[#Space]], though this is more interesting when the dimensions are higher. An example of a Scalar Field on 2D cartesian space is a contour map, it describes the surface of the land by giving the elevation at each place. This surface we walk around in is certainly a curved 2D space, not a euclidean one. We can certainly perceive its slope, see its curvature and consider volume under (or over) --- which becomes rather real when the contours go below sea level and the ocean fills that negative space. | + | Imagine a graph --- the number line with graph above or below it as given by the output of one of the functions in that collection. The usual kind of graph we draw. Now consider that graph as a list of values along the number line. The value 'listed' for each point on the horizontal axis is the height above that axis of the line we drew. Some nice, detailed list of values indeed! With a value given for each point on the number line the dimension is the size of the set of all real numbers. [[#fields|Earlier]] we had thought of such a function as a Scalar [[#Fields|Field]], or thought of a Scalar Field as a function. We can alternatively consider this curved line as a curved one dimensional [[#Space]]. This gets more interesting when the dimensions are higher. An example of a Scalar Field on 2D cartesian space is a contour map, it describes the surface of the land by giving the elevation at each place. This surface we walk around on is certainly a lumpy 2D space, not a euclidean one. We can perceive its slope, see its curvature and consider volume under it (or over). That volume becomes rather physical when the contours go below sea level and the ocean fills that negative space. |
| This way we can start to build intuition, each time we reimagine an abstract something in different ways it gives us a new perspective on those things. Pushing that to a 4D minkowski spacetime which is extended as elastic and with curvature everywhere is our main description of the cosmos. Add torsion, twist, everywhere and perhaps we are approaching the next cosmological description, but leave that to today and tomorrows leading physicists to think about, it is very much one approach to some of the open questions in current physics. Considering the underlying spacetime and the Big Bang one suggestion, certainly not majority view at present but backed by some very interesting calculations from recent detailed observations, is that it is closed in on itself, finite, in an essentially flat hyper-torus kind of structure, expanding and presently maybe only 3 or 4 times longer across than the longest observable distance, distance light could have travelled since the Big Bang. This combines very cleanly with the torsion ideas, and shifts our models on a fairly deep philosophical level and that kind of shift does not happen overnight! But I personally would certainly not bet against it. | This way we can start to build intuition, each time we reimagine an abstract something in different ways it gives us a new perspective on those things. Pushing that to a 4D minkowski spacetime which is extended as elastic and with curvature everywhere is our main description of the cosmos. Add torsion, twist, everywhere and perhaps we are approaching the next cosmological description, but leave that to today and tomorrows leading physicists to think about, it is very much one approach to some of the open questions in current physics. Considering the underlying spacetime and the Big Bang one suggestion, certainly not majority view at present but backed by some very interesting calculations from recent detailed observations, is that it is closed in on itself, finite, in an essentially flat hyper-torus kind of structure, expanding and presently maybe only 3 or 4 times longer across than the longest observable distance, distance light could have travelled since the Big Bang. This combines very cleanly with the torsion ideas, and shifts our models on a fairly deep philosophical level and that kind of shift does not happen overnight! But I personally would certainly not bet against it. | ||
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| In calculus we consider the function, its derivatives and integrals. An interpretation of derivatives and second derivatives is slope and curvature of that line, that is that space. And when we dig deep in the world, going beyond the scales we can touch and see, we must rely on very abstract descriptions, we need to build this intuition about how what we are describing behaves, what it is. If something in the world **is** a function we need to expand how we think about functions. At the atomic scale entities do not have a nice position and trajectory, this fails completely as a description. They are fuzzy, they interact, they merge into larger entities, they are best described as functions. | In calculus we consider the function, its derivatives and integrals. An interpretation of derivatives and second derivatives is slope and curvature of that line, that is that space. And when we dig deep in the world, going beyond the scales we can touch and see, we must rely on very abstract descriptions, we need to build this intuition about how what we are describing behaves, what it is. If something in the world **is** a function we need to expand how we think about functions. At the atomic scale entities do not have a nice position and trajectory, this fails completely as a description. They are fuzzy, they interact, they merge into larger entities, they are best described as functions. | ||
| - | The space in which we exist does not fit a nice three dimensional euclidean space with time flowing independently. That description is a reasonably good approximation as long as we stay very much within the scales and energies we can live in and perceive with our senses. But it does not suffice for developing modern technologies, or describing the cosmos. We exist in a space much more accurately described as a four dimensional spacetime which is dynamically curved and fundamentally elastic. Mass, energy, gravity and this curvature are intimately linked. But that is well beyond school maths, after extension 2 you are in a place to get started, you have some of the language required to talk about all this. You will need a lot more. | + | The space in which we exist does not fit a nice three dimensional euclidean space with time flowing independently, and things looked at closely enough are not made of tactile, moving little objects. That description is a reasonably good approximation as long as we stay very much within the scales and energies we can live in and perceive with our senses. But it does not suffice for developing modern technologies, or for describing the cosmos. We exist in a space much more accurately described as a four dimensional spacetime which is dynamically curved and fundamentally elastic. Mass, energy, gravity and this curvature are intimately linked. But that is well beyond school maths, after extension 2 you are in a place to get started, you have some of the language required to talk about all this. You will need a lot more. |
| This kind of classifying things can be very useful in understanding how they work, and generalising and abstracting them. And building our intuition about structures we cannot just pick up and look at with our human senses. This is getting way outside the scope of high school maths, so I will leave it there. To tackle extension maths means building towards that kind of intuition but quite informally. We do not do a great deal of systematic proving of the properties of the structures we are learning to work with. We do not look at the formal relationships and language that can describe much of what I have spoken of in this discussion much more rigorously. | This kind of classifying things can be very useful in understanding how they work, and generalising and abstracting them. And building our intuition about structures we cannot just pick up and look at with our human senses. This is getting way outside the scope of high school maths, so I will leave it there. To tackle extension maths means building towards that kind of intuition but quite informally. We do not do a great deal of systematic proving of the properties of the structures we are learning to work with. We do not look at the formal relationships and language that can describe much of what I have spoken of in this discussion much more rigorously. | ||
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| An //Isomorphism// is a mapping between structures that preserves significant aspects of their behavior. It can give us a great insight into either form because we have developed insight in each form quite differently. An early example of this is decimals, mixed numerals and improper fractions. We learn to convert between them and each gives us a different insight into the quantity or ratio represented. Each helps us solve different kinds of problems. The process of drawing a graph of an equation is an Isomorphism. Two structures can be //Isomorphic//. In this case the equation and the graph both show the same underlying relationship between the values of `x` and `y`. For some questions about that relationship an equation and the algebra we know about equations will give us what we need. For other questions about the relationship the shape of the graph and what we know about geometry will be a much more direct answer. More likely in the context of our school program is that the graph will give us essential insights into the problem which can then be followed up with techniques we have learnt about functions. | An //Isomorphism// is a mapping between structures that preserves significant aspects of their behavior. It can give us a great insight into either form because we have developed insight in each form quite differently. An early example of this is decimals, mixed numerals and improper fractions. We learn to convert between them and each gives us a different insight into the quantity or ratio represented. Each helps us solve different kinds of problems. The process of drawing a graph of an equation is an Isomorphism. Two structures can be //Isomorphic//. In this case the equation and the graph both show the same underlying relationship between the values of `x` and `y`. For some questions about that relationship an equation and the algebra we know about equations will give us what we need. For other questions about the relationship the shape of the graph and what we know about geometry will be a much more direct answer. More likely in the context of our school program is that the graph will give us essential insights into the problem which can then be followed up with techniques we have learnt about functions. | ||
| - | A graph is not much help understanding the relationship it describes if you do not have strong geometric insight. Developing that insight is the main goal of the //Measurement and space// thread that has been in the syllabus since kindy. Recently renamed --- it had been called Measurement and Geometry. Geometry was demoted below the core threads in 2024, but remains in many content areas. Geometrical construction has been dropped, it seems, as explicit content. But geometrical reasoning most certainly has not been, and to teach and develop it surely requires understanding constructions. So it is still in there. Without understanding, problem solving, reasoning, communications and fluency in these remaining content areas there is little geometrical insight, half of 'relationships' is missing. They are vital elements of the syllabus and must be tackled and tested seriously. Equations and functions are part of the //Number and algebra// thread, also since kindy. Again the main goal here is understanding mathematical descriptions of quantity of the numerical kind. The power of mathematical reasoning and problem solving is using very different insights into a situation enabled by multiple different models of it. | + | A graph is not much help understanding the relationship it describes if you do not have strong geometric insight. Developing that insight is the main goal of the //Measurement and space// thread that has been in the syllabus since kindy. Recently renamed --- it had been called Measurement and Geometry. Geometry was demoted below the core threads in 2024, but remains in many content areas. Geometrical construction has been dropped, it seems, as explicit content. But geometrical reasoning most certainly has not been, and to teach and develop it surely requires understanding constructions. So it is still in there. Without understanding, problem solving, reasoning, communications and fluency in these remaining content areas there is little geometrical insight, half of 'relationships' is missing. They are vital elements of the syllabus and must be tackled and tested seriously. Equations and [[#Functions]] are part of the //Number and algebra// thread, also since kindy. Again the main goal here is understanding mathematical descriptions of quantity of the numerical kind. The power of mathematical reasoning and problem solving is using very different insights into a situation enabled by multiple different models of it. |
| ===geometry=== | ===geometry=== | ||
| - | The kit you brought to the beginning of high school, assuming you did year 7 before 2024, almost certainly included a box with compass, dividers, protractor, a ruler and two triangles. Unless you were very lucky and had great teachers you probably hardly touched it. That part of the syllabus was usually very badly neglected. Somehow proper geometric construction on paper never quite got into the classroom. It could be replaced by geometrical construction on a computer, but is not in practice. A ruler to measure distances occasionally (but mostly make your working look neat), a couple of lessons were you had to measure angles with the protractor (that is still in content) and maybe drawing a few circles with the compass (mainly to make your diagrams pretty). If pencil and paper is too old fashioned you __must__ fully embrace digital constructions in the mathematics classroom and in homework, which in many ways is more sophisticated while demanding the same kind of practical understanding of spatial relationships. In the digital realm the connection between number and geometry is much more explicit. There are Computer Aided Design programs that everybody now uses rather than a pencil --- but with the real ones you do constructions the same way, drawing points, lines, angles, circles, intersections and such. You must understand the constructions and have, or develop, strong geometrical intuitions using them. Maybe you have independently done some design or computer programming that involves computer graphics. Maybe you happened to get interested in some older books of mathematical puzzles, there are a great many of those still around. Maybe you get out saws and drills, or a 3D printer, from time to time and build something. | + | The kit you brought to the beginning of high school, assuming you did year 7 before 2024, almost certainly included a box with compass, dividers, protractor, a ruler and two triangles. Unless you were very lucky and had great teachers you probably hardly touched it. That part of the syllabus was usually very badly neglected. Somehow proper geometric construction on paper never quite got into the classroom. It could be replaced by geometrical construction on a computer, but is not in practice. The ruler to measure distances occasionally (but mostly make your working look neat), a couple of lessons were you had to measure angles with the protractor (that is still in content) and maybe drawing a few circles with the compass (mainly to make your diagrams pretty). If pencil and paper is too old fashioned you __must__ fully embrace digital constructions in the mathematics classroom and in homework, which in many ways is more sophisticated while demanding the same kind of practical understanding of spatial relationships. In the digital realm the connection between number and geometry is much more explicit. There are Computer Aided Design programs that everybody now uses rather than a pencil --- but with the real ones you do constructions the same way, drawing points, lines, angles, circles, intersections and such. You must understand the constructions and have, or develop, strong geometrical intuitions using them. Maybe you have independently done some design or computer programming that involves computer graphics. Maybe you happened to get interested in some older books of mathematical puzzles, there are a great many of those still around. Maybe you get out saws and drills, or a 3D printer, from time to time and build something. |
| Otherwise you have a lot of geometrical intuition building to catch up on! You do need it. It is expected in the exam questions, it is part of the maths syllabus. It is vital in many real world applications and to developing spatial understanding. | Otherwise you have a lot of geometrical intuition building to catch up on! You do need it. It is expected in the exam questions, it is part of the maths syllabus. It is vital in many real world applications and to developing spatial understanding. | ||
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| In this discussion we have considered Vectors as 'magnitude with direction' and as 'a list of scalars', the second is the Cartesian form. We sometimes call these the 'Polar' and 'Rectangular' forms. They are different ways of writing and saying the same underlying value. As with finite decimals and fractions. We learn to convert between them. This conversion is another isomorphism, and each form gives us different insights and different possibilities for calculations. One is algebraic, one is geometric. The power comes from the combination, not 'oh goody I can get an answer using numbers instead!'. | In this discussion we have considered Vectors as 'magnitude with direction' and as 'a list of scalars', the second is the Cartesian form. We sometimes call these the 'Polar' and 'Rectangular' forms. They are different ways of writing and saying the same underlying value. As with finite decimals and fractions. We learn to convert between them. This conversion is another isomorphism, and each form gives us different insights and different possibilities for calculations. One is algebraic, one is geometric. The power comes from the combination, not 'oh goody I can get an answer using numbers instead!'. | ||
| - | A scatter graph is isomorphic with a 2-column matrix showing two distinct aspects measured for many examples. It shows that data quite clearly, and can help us 'see' that matrix and the data it represents. We can find or describe patterns in that data, patterns which may have been difficult to 'see' in the 2 column table of data. Kepler, Dirac and a few other very unique minds are able to see very complex and surprising relationships in raw data and break very new ground indeed. For the rest of us an equation and a graph helps a great deal --- and calculus when we are trying to understand changes in something. The whole language of graphs, calculus and functions was invented (in that order) to explain and extend Kepler's insights, they were not available to him at the time. They had studied formal geometry very closely. The algebraic formulations came next. Again equations and graphs, one algebraic one geometric. | + | A scatter graph is isomorphic with a 2-column matrix showing two distinct aspects measured for many examples. It shows that data quite clearly, and can help us 'see' that matrix and the data it represents. We can find or describe patterns in that data, patterns which may have been difficult to 'see' in the 2 column table of data. Kepler, Dirac and a few other very unique minds are able to see very complex and surprising relationships in raw data and break very new ground indeed. For the rest of us an equation and a graph helps a great deal --- and calculus when we are trying to understand changes in something. The whole language of graphs, calculus and [[#functions]] was invented (in that order) to explain and extend Kepler's insights, they were not available to him at the time. They had studied formal geometry very closely. The algebraic formulations came next. Again equations and graphs, one algebraic one geometric. |
| ===Science=== | ===Science=== | ||
| The modern description of the universe, and just about everything in it, is pretty close to pure equations. Entities are blurry geometrically, they are equations that interact. Space, time and spacetime is not some kind of simple cartesian space that they, and we, exist in. It is warped and springy, and time is not what it seems. We understand space, time and entities by the various equations that describe them. Alternative equations give different insights. Geometric interpretations of the equations are critical. Algebraic and geometric intuition must both be pushed very hard here. | The modern description of the universe, and just about everything in it, is pretty close to pure equations. Entities are blurry geometrically, they are equations that interact. Space, time and spacetime is not some kind of simple cartesian space that they, and we, exist in. It is warped and springy, and time is not what it seems. We understand space, time and entities by the various equations that describe them. Alternative equations give different insights. Geometric interpretations of the equations are critical. Algebraic and geometric intuition must both be pushed very hard here. | ||
| - | The philosophical and scientific journey from Galileo (Infinitesimals) to Euler (Functions and Complex Numbers) was the context for Kepler's fabulous insights. It took about a century and is a very big part of the maths we teach in high school. The journey into Pure Maths as we understand it now came next, getting started maybe 300 years ago. That is towards a formal understanding of what Maths is, making the language and ideas rigorous, seeing the structures of structures, digging down to deeper foundations (they are endless, there is __no__ bottom) and up to broader abstractions and generalisations. This is the language required for modern science, analysis, computation and planning. We do not go there in high school. We do look at applications of calculus and statistics in ways that developed alongside that Pure Maths journey. We hope to put a few students into a place they could be comfortable starting to follow that journey during university maths. Maybe then to participate in the scientific, analytical, engineering or computation worlds that have come from that. Dirac's incredible insight was early in the shift from classical dynamics to the relativity and quantum mechanics understanding of the world, the shift to much more pure equations. | + | The philosophical and scientific journey from Galileo (Infinitesimals) to Euler ([[#Functions]] and [[#Complex Numbers]]) was the context for Kepler's fabulous insights. It took about a century and is a very big part of the maths we teach in high school. The journey into Pure Maths as we understand it now came next, getting started maybe 300 years ago. That is towards a formal understanding of what Maths is, making the language and ideas rigorous, seeing the structures of structures, digging down to deeper foundations (they are endless, there is __no__ bottom) and up to broader abstractions and generalisations. This is the language required for modern science, analysis, computation and planning. We do not go there in high school. We do look at applications of calculus and statistics in ways that developed alongside that Pure Maths journey. We hope to put a few students into a place they could be comfortable starting to follow that journey during university maths. Maybe then to participate in the scientific, analytical, engineering or computation worlds that have come from that. Dirac's incredible insight was early in the shift from classical dynamics to the relativity and quantum mechanics understanding of the world, the shift to much more pure equations. |
| Going back to that 2 column data above we might be able to find an equation that fits the relationship between `x` and `y` quite closely and that would make a much more concise description of the data. It might help us understand the nature of the relationship much more deeply because we have become familiar with the nature certain equations, and those insights can be applied to any relationship described by those equations. We aim to become very familiar with linear, quadratic, exponential and trigonometric relationships especially, and if we get there we can nail lots of problems in the HSC exams. And they all turn up everywhere in nature, technology, economics and planning. | Going back to that 2 column data above we might be able to find an equation that fits the relationship between `x` and `y` quite closely and that would make a much more concise description of the data. It might help us understand the nature of the relationship much more deeply because we have become familiar with the nature certain equations, and those insights can be applied to any relationship described by those equations. We aim to become very familiar with linear, quadratic, exponential and trigonometric relationships especially, and if we get there we can nail lots of problems in the HSC exams. And they all turn up everywhere in nature, technology, economics and planning. | ||
| Throughout school we have learnt procedures where we can represent something in a different way, with different notation or in a quite distinct form. Then we use methods we have learnt in one representation to solve problems presented in another way. But we do not think too much about what we are actually doing. Though I certainly try to make it clear what this flipping between forms is, that these forms are equivalent but different representations of the same thing, I never use the term Isomorphic --- it is a bit intimidating and following it up (something I hope students might do occasionally after our discussions!) you would get into all manner of very formal pure maths of a kind very different to what we look at in school. That is for uni, but only if you choose the Pure Maths rather than the stats or analysis paths. | Throughout school we have learnt procedures where we can represent something in a different way, with different notation or in a quite distinct form. Then we use methods we have learnt in one representation to solve problems presented in another way. But we do not think too much about what we are actually doing. Though I certainly try to make it clear what this flipping between forms is, that these forms are equivalent but different representations of the same thing, I never use the term Isomorphic --- it is a bit intimidating and following it up (something I hope students might do occasionally after our discussions!) you would get into all manner of very formal pure maths of a kind very different to what we look at in school. That is for uni, but only if you choose the Pure Maths rather than the stats or analysis paths. | ||
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