I’ve been working on presenting Matrices in a way that really extends and puts in context some ideas that are rather big in the HSC. Here I am trying to show a few of those big ideas from quite a different perspective, and I hope this really helps your thinking about them — and your ability to apply them in problem solving!! This is growing by the hour, it has been for a few weeks, after a summer going seriously down the quantum rabbit hole and landing on matrices. They really really should be part of extension HSC, they are deeply relevant for a lot of modern physics and now even more seriously important in computing and making sense of a whole range of very current discussions.

Read these paragraphs slowly and think about these perspectives on a lot of the ideas you are studying in extension HSC. Think specially about how these areas are tied together. That is a key to being able to understand and solve problems that involve the methods you have memorised and practiced in class. And solving those problems is the goal of the course, and a huge part of the exam.

I want to add a few words to your mathematics vocabulary. Those headings and bold-face words here that are new, or that are more broadly defined here. They are about ideas you have tackled at school but have not named. Naming something certainly helps thinking about it. The topic Ext2 is about to hit in the first 5 weeks of year 12 — Mathematical Proof — is rather close to the Pure Maths perspective that underlies the whole of this discussion. The kind of insights I end up talking of in Isomorphism below are vital for the kinds of creative proofs that form the toughest questions in Ext2.

But that understanding, putting it all together as one big language is also required, perhaps less creatively and in more familiar contexts, in Ext1 problems and in the tough questions in Advanced.

These are like tables of numbers (or some other thing we can add and multiply in the usual fashion), but most commonly we talk of Real Numbers. When we move beyond real numbers the ideas can get very interesting, and very useful. They are similar to vectors, which we can write as a list or column, also often but not always Real Numbers. We can add matrices and multiply them by a scalar in the same way we can with vectors written this way. This multiplying by scalars and adding is called Linear Combination and is a very important idea in many contexts. We have other operations we can do with them — which makes a Matrix Algebra.

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. 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 work towards in this discussion.

This essentially means a value we can use to multiply something, or to measure a single aspect of something. The magnitude of a vector is a scalar, so are the elements of matrices. Multiplying by a scalar often relates to stretching or shrinking something — changing its scale. It certainly does when we are talking of measurements and Real Numbers, or Real Vectors. Sometimes we think of matrices as a list of vectors, so the rows of the table are vectors, or maybe the columns are vectors. Each vector in the list must be the same type – all the same dimension (have the same length list of scalars) and use the same kind of scalar. A scalar always has a Unit, multiplying by that value leaves the thing unchanged, it neither stretches nor shrinks. We usually use `1` as its symbol and call it the Identity.

We meet them in high school. At first they simply have a magnitude and a direction. We can use an arrow (its length is the magnitude) in a space to show one. The magnitude is a scalar. We can multiply vectors by these scalars. This stretches or shrinks them while preserving direction, reversing it if we multiply by a negative. We can also add vectors. The negative of a vector is just ‘in the the opposite direction’. Being negatives of each other means they add together to make zero and that adding one of them undoes the addition of the other. In this addition sense vectors are very like numbers. When the magnitude is `1` then we have some sense of a basic direction, we call that a Unit Vector. This becomes extremely important and powerful as we bring together geometry, number and algebra. We can move these arrows around in geometrical space, it does not matter — the magnitude and direction stay the same, it is the same vector everywhere. The arrow represents a value, familiar ideas like velocity have this kind of value and velocity is the same no matter where you are (at least until you get into modern physics and things get much more complicated!! Hmm — Matrices).

To add vectors we can put the base of the first arrow on a place we call `0` then put the next arrow on the tip of the first — and the sum is an arrow from `0` to the tip of the second. This is rather like jumping around on a number line, but we are not confined to a line. We can also think differently about our vectors. Put the arrow on `0` and put in an axis through that `0` for each dimension using our scalars as the scale — then look at where the tip of the arrow is in this Cartesian Space. We describe position with a list of numbers, one for each dimension. This description is an example of what we called a vector above, when we were talking about matrices.

A list of scalars or a magnitude and direction are both descriptions of a vector. We can do our vector operations in the list of scalars form, they are equivalent. They must be the same type, in the same space — that is have the same dimension with the same kind of scalar. We add each corresponding pair from their lists of scalars individually. We can multiply a vector by a scalar — multiply each scalar in the list individually. These are equivalent to the addition and scalar multiplication above.

There are some other vector operations. One that will be important in this discussion is the Dot Product, also called Scalar Product because the result is a scalar not a vector. This is between two vectors of the same type — the same dimension with the same scalars. Multiply each pair of corresponding elements individually then add these up. We often use bold face for vectors and the same letter with an index for its scalar components, so for `n`-dimensional vectors: \(\ {\bf v\bullet w}=\sum_{i=1}^n v_i w_i.\ \) Or if we are thinking magnitude and direction, with `|\ |` indicating ‘magnitude of’ and \(\ \theta\ \) the angle between them: \(\ {\bf v\bullet w}=|{\bf v}| |{\bf w}|\cos\theta\ \) is equivalent.

This makes for some useful calculations. In dynamics for example \(\texttt{force}\bullet\texttt{displacement}\) is work done, which is energy transfered. Considering movement along a curve with a varying force we can add up the infinitesimal contributions, \(\ \int{\bf F}\bullet{\rm d}{\bf s}\ \) but calculus with vectors or multiple variables is just beyond where we get at school so we often stay with constant \(\bf F\) and straight tracks. Dynamics questions are in our maths course because they provide some easy to visualise examples of mathematical models applied to the world. If the vectors are perpendicular dot product will be `0` since \(\ \cos\frac\pi2=0,\ \) if they are parallel it will be the product of their magnitudes. This is closely related to the idea of independent components. If everything is all on the same straight line our model is 1D, only the magnitudes count and direction is simply positive or negative. Work done (potential energy added) on a load lifting it in an elevator or with a crane is an example of this. The two quantities are parallel, we can treat them as scalars.

If one of the vectors is a Unit Vector we get the scalar component of the other vector in that direction. But more usually we have a pair of vectors and we want to know the component of \(\bf v\) in the direction of \(\bf w\). In our maths course this often comes up in projectile motion. Here we have a force (hence acceleration) aways in the same direction so it only changes that component of the velocity, we consider the components of the velocity independently and put them back together in the end. This kind of analysis can be applied in many different situations. Divide a vector by its magnitude and it becomes a Unit Vector, so the component is \(\frac1{|{\bf w}|}{\bf w\bullet v}.\) With this magnitude the perpendicular component is the other leg of a right angle triangle and pythagoras rule gives that easily. Notice that Dot Product is Associative and Commutative along with scalar multiplication and scalar division is multiplication by reciprocal as usual. We can rearrange our vectors like we do variables in an algebraic Term. All this is especially useful if we have 3D (or higher!) vectors in cartesian form, Pythagoras gives us \(|{\bf w}|\) easily but the angle is a bit messy geometrically. Dot product finds angles between 3D vectors quickly, putting the magnitude-direction form and cartesian form together we get \[\sum_{i=1}^n v_i w_i=|{\bf v}| |{\bf w}|\cos\theta.\] Make that into a formula if you like but these should not be memorised formula for particular questions!

Be familiar with the two forms of vectors, the dot product in each form, its interpretation for component, angle or ‘are they perpendicular?’ and pythagoras for magnitudes. Be practiced noticing complementary or supplementary angles, right, isosceles and equilateral triangles, angle sums and all those geometrical features. Be familiar with noticing when these direct calculations or relationships could be part of the result you are seeking, the information you are given or some intermediate step. The problems in the exam are devised to test your imagination and understanding of the relationship between models and the calculations you can use in them.

Describe the relationship between the vectors `[2, 4, 1]` and \([5, -3, 2]\).

What is the area of the shape with vertices \([3, 2, 1], [5, 6, 2], [8, -1, 3]\) and `[10, 3, 4]`

Try finding the angle between `[4, 1, 3]` and `[6, 2, 1]`.

Without using a calculator find the angle between `[2, 10]` and `[3, 2]`. There are at least a couple of very different ways to tackle this.

Geometrically using right triangles and trig these are messy, with Dot Product easy to write down what needs to go in the calculator. It may be quicker just to use the coordinates, square roots and such in a big fraction but the school calculator models do have vector mode.

Unless you are doing several calculations with the same few vectors fussing around with those menus and the absence of a history in that mode probably outweighs any advantage you get with vector operations. Polar–Rectangular conversion is available in the standard mode, for 2D vectors.

Try:
A train of mass \(m\) is rolling back on a straight track with speed \(a\). A force with given magnitude, \(f,\) is applied to the train at angle \(\theta\) to the track, it stops the train. How far back on the track does it come to rest.

We could certainly draw a diagram and find right angle triangles, use our trig and find the appropriate intermediate values then come up with the answer. There are many ways to tackle the problem. It is a very well known type of system and there are a lot of standard formula that can give a string of intermediate results and then a final outcome. But maybe we can use our vector knowledge and calculus to express the problem in a way that could generalise to much more complicated systems. In high school we look at highly simplified problems but we take away the usual computer maths support and expect them to be analysed using the tools we are teaching. Walk before you run! But really build the language from ground up. The HSC questions are very carefully structured to test understanding and flexibility when using the tools.

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 in this discussion.

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 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 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.

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 a new function that gives a meaningful and useful answer to the question “What is the slope of this space?” and a way of going backwards on this that turns out to answer a very different question about changes as well as “How much area is there under a part of this space?”. Another way to say these are “How fast are things changing at a particular time?” (Derivative) and “What is the total change during an interval of time?” (Definite Integral). 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. The new understanding of the world enabled the industrial revolution, and much of what we think of as a comfortable life.

We also talk about the set of all the vectors of a particular type and dimension along with their scalars. We call this collection a Vector Space. This set is in many ways like our various sets of numbers of a particular kind, its structure and behavior has much in common with the structure and behavior of these numbers. It also has much in common with the structure of Euclidian Space considered as a set of points.

The mathematical power here is in bringing the number properties addition, subtraction and difference together with the vector properties of scalar multiplication and dot product and considering them as working with the points in geometric space. That is quite a mouthful of words. That is why we take a few years and several big school topics working with different aspects of this one sentence — finally hitting Vectors and (for Ext2) Complex Numbers.

We define the structure Vector Space by making addition follow the rules of number addition. We call any set with such an operation a commutative Group, an Abelian Group. Then we define scalar multiplication to extend that by following a hybrid version of the rules of number multiplication, between one of each type of value. We call integers (or any other set with \(\ +\!\times\!=\ \) of its kind) a Ring. With a Vector Space we have two kinds of value (vector and scalar). The scalar must be more than just a Ring, it must support full division like Rational Numbers do. The vectors do not have this number kind of multiplication between them, Dot Product gives a scalar not a vector. Vectors gain a direction property which can also be interpreted as the set being `n`-dimensional. That means they lose the sense of order that many types of number have. With Real Numbers \(\ne\) implies either \(\lt\) or \(\gt\). This is not true for vectors. Order is essentially a one-dimensional quality.

Square matrices do have multiplication between them, and thus they have a Unit, an Identity and much else, but that multiplication is non-commutative, and only works between particular combinations of non-square matrices, More on that here.

By dimension above we essentially mean the number of independent directions our set of vectors has. We can start with a carefully chosen set of vectors which we call a Basis and generate all the vectors by the Linear Combinations of them. Put another way you can make every vector by a scalar multiplication of each basis vector then adding the results together. A basis must be one of the possible smallest sized sets of the vectors that can do this. In which case there is only one way to do it for each vector. When we were considering vectors and put them into a Cartesian Space above we effectively chose a basis. We chose the Unit Vectors along each axis. The dimension of a Vector Space is the number of vectors in its basis, the dimension of a Cartesian Space is the number of axes. The values of the components of a vector depend on the basis chosen, those operations using the scalar components are assuming we are using the same basis consistently. Suppose we have chosen a basis of Unit Vectors \(\ \{{\bf a, b, c}\}\ \) that means every vector in the space is uniquely represented by three scalars as \(\ [x\ y\ z]\ \) such that \(\ {\bf v}=x{\bf a}+y{\bf b}+z{\bf c}.\ \) This is a Linear Combination of the basis and \(\ x, y, z\ \) are the components of the vector, or the coordinates of its tip if it is placed on point \(\ (0,0,0)\ \) in the corresponding Cartesian Space (the axes use the same scalar and are in the same direction as the basis vectors). The idea of Linear Independence is important here, a vector is independent of a group of vectors if there is no Linear Combination of them that makes that vector.

When studying vectors seriously we prove that all those equivalences, uniquenesses, properties and the idea of a basis are valid and consistent. This does give us a much deeper insight into how it all fits together, but we only get started with this idea of Proof in high school, that kind of detailed analysis of mathematical structures must wait. But we must get a general intuition into the nature of the mathematical structures we use, and the relationships between them. This is the way we use them to solve problems, and the way we know how to tackle the HSC questions. Those questions are carefully devised to test if you have developed that intuitive understanding!

There is an important sense in which Scalars, Vectors, Matrices and Spaces have overlapping representations. I have already mentioned that a Vector can be considered a single column or single row Matrix, and we saw how Vectors corresponded to Cartesian Space of the same dimension. Consider a scalar, \(s,\) as the one-element matrix \(S=\left[s\right]\) and as a 1D vector \(\bf s\) with magnitude `|s|`. (In a single dimension direction is simple: positive or negative.) \(\bf s\) corresponds to a point on a line, distance `|s|` in the positive or negative direction from `0`, the same as `s` on the number line. 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 a single entry matrix result is considered a scalar.

Look here for more detail on these matrix operations. 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. They become very important in transformations within spaces, or mapping between spaces. They become deeply part of the language of physics. They are fundamental to computations in many areas. As mathematical language they have many applications.

Now consider square matrices. For matrix multiplication with `n`-dimenional square matrices we have an Identity, the Unit for multiplication, like `1` in the scalars, \[I=\left[\matrix{ 1&0&\cdots&0\cr 0&1& &0\cr \vdots& &\ddots& \cr 0&0& &1\cr }\right]\ \text{ because }\ IA=AI=A=1A.\] Consider: ` X=xI=[ [x,0],[0,x] ].` `X` works like `x` with both 2\({\times}\)2 matrices 2D vectors. Consider examples \(A, V, \bf v\) where `V` is \(\bf v\) written as single column matrix. \(\ xA=XA=AX\ \) and \(x\bf v\) corresponds to \(XV\).

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!

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.

The reason I dove into this quantum computing rabbit hole and landed here on matrices is because these can be very hard computations which quantum computing can tackle in an entirely different way. And that wave is just breaking, especially in the last few months. Many calculations essentially impossible classically are starting to become quite reasonable within a Quantum Computer and they have become usefully operational in the last few months, beyond a few very specialist applications that have been around for a few years. They are rapidly scaling up towards doing what was very recently impossible. Neural Nets are another computational approach that has already shattered old limitations. Work 40 years ago that ultimately led to them as today’s very effective mathematical and computational tool was given the physics Nobel Prize last year, and applications were given the Chemistry Nobel Prize. They are also known as AI. Both Quantum and Neural Net calculations are profoundly mathematical approaches that are only comprehensible as pure mathematical structures, and matrices are deeply part of that.

At quantum level reality is pure equations, matrices and complex numbers are very real and to use that for computation we look for how we can turn a problem into equations that we can set up across a collection of entangled electrons or photons or structures made with them and watch them resolve the question almost instantly – then read out the answer. That sounds complicated, but is actually a gross oversimplification. This stuff truly is hard and counter intuitive, and it is pure maths that gives the handles to help the imagination get there and shift our intuition into this very different scale and world.

We get as far as them in high school, if you go to our top level of HSC maths. There are lots of simple algebraic equations that have no solutions in Real Numbers. Consider \(x^2+1=0.\) Nothing fancy or tricky, but no solution for `x`. But simply invent one more number, the ‘positive’ solution to this equation, give it the name \(\rm i\), then include all the numbers you get by adding and multiplying it with the rest of the Real Numbers. Essentially every number \(\ a+b\rm i\ \) with `a, b` real numbers and \(\ \rm i^2=-1.\ \) Now all our polynomials, and much else, always have solutions. And it is entirely consistent, these numbers obey all the usual number rules and we can easily use them in number algebra. They can multiply things like matrices or vectors easily enough if you allow them as the numbers in the table, list or magnitude. They can be the coordinates of a space. It is hard to imagine them as a size, there are very real problems here, and they do not have a place on our familiar number line — so when they are scalars we need to do some serious rethinking. Complex quantities, Complex Vectors, Complex Cartesian Space, Complex Fields and Complex Matrices have very natural applications to the physical world. The geometrical visualisations of these concepts is challenging, we are very accustomed to thinking with a Real Euclidian Space set in a flow of independent continuous time. The world is not so simple once we get big, small, energetic or cold enough and to understand it, work with it, predict it or make and use modern technologies we have to really shake up that habit. We need these mathematical structures as the language to do it. There are important operations with them that do result in Real Numbers, and these can give various senses of size but not really order because many share the same size. It turns out nature is quite happy with them, there are quantities out there that make a lot of sense as Complex Numbers. Again we have some serious new intuition building required if we are to imagine, predict and describe nature as we find it when we look closely, and that intuition will have to come from serious pure mathematical thinking. The science of the 20th century was a spectacular example of doing exactly that, and the 21st is looking wilder still.

Considered as numbers they have a two dimensional representation, we have a Complex Number Plane that is like the Real Number Line. We put \(\rm i\) directly above `0` on the number line. Same distance up as 1 is to the right. The rest all works, \(a+b\rm i\) is at the coordinates \( (a,b) \) in this plane. But keep going … that means they have a ‘distance’ from `0` and a direction, the angle measured anticlockwise from `1` to the point in this Complex Number Plane. In this respect they are like 2D Vectors. But they are much more — you can multiply and divide them, take their log, and a great deal more that you cannot do with Vectors. They behave like both vectors and numbers, adding is the same as adding vectors and multiplying by a real number is the same as with Real Vectors. In high school we find that we can raise a number to a complex power quite sensibly and consistently. And with this there are some very powerful connections made between parts of maths that till now had seemed quite distinct. Algebra, Geometry and Trigonometry come together in Complex Numbers.

Some consider the equation that summarises this idea, \(\ \rm e^{i\pi}+1=0,\ \) the most remarkable in mathematics. It connects each of the five basic numbers and three basic number operations. But more on it soon, it has much to do with matrices and physics — and quantum computing.

Complex numbers are only in extension 2 high school here, it ties together a lot of the ideas you have been developing over the earlier topics. There is at least one fundamental matrix algebra idea that I very much should share with you that is part of this. The relationship between matrices and complex numbers is something I had not really explored before, and is very elegant as well as deeply core to the way the world works on the basic entity level, electrons, photons, atoms and such. But a representation of complex numbers (and hence also real numbers) within the 2\(\times\)2 square real matrices is intriguing. I’ll need to cover a bit about such matrices first though!

Real Vectors, with Real Number magnitudes and finite positive integer dimension usually 2 or 3, thought of with direction in geometric space are the kind we look at in high school — like displacements, forces, velocities or accelerations. But beyond that school topic we can have Complex Vectors, where the magnitude is a complex value. Thinking about arrows in geometric space breaks down here, but Nature does not care! But that is just the start. We can consider polynomials as vectors (the coefficients are the list of scalars). 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! Consider the simplest case, a function from Reals to Reals. Imagine the graph, a number line with graph above or below it as given by the function output. The usual kind of graph we draw. Now consider that graph as a list of values along the number line. A very long and dense list, thats for sure! But that is an infinitely long list of real numbers. An infinite dimensional vector. And functions do behave like vectors in this way. It can be a very useful way of generalising and understanding functions as abstract entities.

In school we met 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. Its direction is perpendicular to each and up or down according to the Right Hand Rule. It is an area and a direction, an oriented area. 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. In higher dimensions we can have an `n`-vector, for example a 3-vector has a volume as its magnitude. But this is again going far beyond school, we limit our Cross Product exploration to few useful applications of in three dimensions.

Consider our familiar functions like \({\rm f}(x)\) giving a real number for each real number in a particular domain. The Vector Space which has all the possible examples of such a function has infinite dimension, and not just the infinite of the integers. That would be an endless list of values one after the other. But this is the much bigger infinite of the number of real numbers or the number of points on a line. These vectors have an independent direction for every point on the number line, whatever that might mean. If we wanted to imagine writing it out as a list of scalars then that list would have an item at every point on a line. Quite a crowded list! But when we draw a graph we do just that. 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! A value for each point on the number line, the dimension is the size of the set of all real numbers. Earlier we had thought of such a function as a Scalar Field, or thought of a Scalar Field as a function. 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. 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.

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.

But it is useful to remember that we can consistently add pure functions and multiply them by a scalar, and reasoning this way is very useful understanding what is happening when we differentiate more complicated functions and we certainly do this in high school.

This is an idea that crops up in many contexts. Starting from Linear Relationships which was a big topic year 9 in school. That is the kind of relationship between 2 values that shows up as a straight line on a graph in cartesian space with those values as the axes. Then extended to many variables.

The equation \(\ ax+by+cz=d\ \) describes a linear relationship between `x`, `y` and `z` when `a`, `b`, `c` and `d` are constants. This relationship could be shown as a plane in a 3D graph. A Linear Relationship graphs as a flat shape one dimension smaller than the number of variables. Some of the constants might be 0, you will still get a plane in 3D space (or nothing or everywhere if \(a=b=c=0\!\) ).

\(\ ax+by+cz\ \) is a Linear Combination of three variables `x, y` and `z`. Much more generally a Linear Combination of a set of things is made by multiplying each by a scalar (possibly zero or one) and adding the results. If there are `n` elements in the set `X` then we could write this \(\ \sum_{i=1}^n a_i x_i\ \) where `a_i` are scalars and `x_i` are the elements of `X`.

We sometimes use the name Linear Space rather than Vector Space. It refers to the kind of independence each dimension of the space has. Any set of Basis vectors must be Linearly Independent, that is none of them are a Linear Combination of the others. This is closely related to Dimension or the Degrees of Freedom within some model. Variables are rather like Degrees of Freedom, you can adjust each quite independently of the others. The space in which the relationship above was graphed is 3-dimensional, you need to pick three values to get to any point. The relationship given could be considered a function of `x` and `y` by putting `z` as the subject, defined as \(\ {\rm f}(x,y)=-\frac1c(ax+by-d)\ \). Now we see there are two variables input with an output for each possible input pair. A way of considering the graph of it is \({\rm f}(x,y)\) as the height above (or negative, below) a cartesian plane with axes `x` and `y`. It will be the same surface in 3D space as the equation gave, but we no longer have any mention of `z`. It is a 2D surface because it is the output of a function with 2 Degrees of Freedom, two variables that may be independently varied. It is a flat surface, a plane, because \(\ ax+by\ \) is a linear combination of `x` and `y` (which is then subtracted from `d` then scaled by \(\frac1c\!\) ).

This idea of independence , dimension and degrees of freedom is deeply part of Vector Space.

Matrices have a great deal to do with that.

is from space to space preserving addition and scalar multiplication. By this we mean we can do the transformation then the operation or do the operation first then do the transformation and the result will be the same. \[{\rm f}:V\mapsto W\quad\text{is linear iff}\quad{\rm f}({\bf a}+{\bf b})={\rm f}({\bf a})+{\rm f}({\bf b})\wedge{\rm f}(c{\bf a})=c{\rm f}({\bf a}),\] so linear combinations are preserved (since addition and scalar multiplication each associative).

eg \(\ {\bf v}\mapsto c{\bf v}\ \) but not \(\ x\mapsto x^2\ \) or \(\ x\mapsto x+1.\) The last is affine instead. For a cartesian or vector space rotation, reflection, stretching or skewing preserving the origin are linear maps, linear transformations. For the appropriate space of functions differentiation, definite integral and indefinite integral with a fixed starting point are linear maps, linear operations on those functions. Expected value of a random variable is a linear function (variance is not). I’ve seen HSC questions that are very quickly answered if you understand the meaning of this last sentence, though most students just use particular memorised properties from the ‘expected value’ topic — properties that are actually true of any linear function, and much much more useful than that single, specific, topic. Many probably could not name or use the much more general and widespread idea and think of the topic in isolation as one more group of methods to memorise.

We can use matrix algebra and notation to work with transformations including linear and affine ones.

This is something we use all the time in school problem solving, without ever naming it.

It is fundamental to mathematical methods and thinking. The word and idea is part of formal thinking and reasoning about mathematical structures, for example about the relationship between \(\ y={\rm f}(x)\ \) and a graph of that equation on the cartesian plane with `x` and `y` on the axes.

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 and which can be followed up with techniques we have learnt about functions.

School classrooms have pulled back from early formal geometrical investigations. They are vital in many real world applications. Somehow proper geometric construction on paper never quite gets into the classroom. Neither does any serious technical drawing on a computer (at least in a context thinking mathematically about construction rather than just copy-paste some code and fiddle till it works). These are the alternatives but usually the first exposure to this kind of mathematical thinking is a few proofs. This is jumping in the deep end to learn to swim — most sink, some quickly realise they can flap their arms and legs around and effectively keep their head above water. For this reason very few test questions actually require a proof — teachers and parents do not like lots of blank or just plain wrong answers. But these questions are in the HSC, and the geometrical insight they depend on is part of many of those questions. It has been part of the syllabus, in gradually more sophisticated ways, every year since early primary. Most classrooms sadly fail to develop it.

The kit you brought to the beginning of high school 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 is often very badly neglected. 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 and maybe drawing a few circles with the compass (mainly to make your diagrams pretty). 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 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. Otherwise you probably 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.

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. This time one is algebraic, one is 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.

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.

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.