structures:matrix:linear
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| so linear combinations are preserved (since addition and scalar multiplication each associative). | so linear combinations are preserved (since addition and scalar multiplication each associative). | ||
| - | eg \(\ {\bf v}\mapsto c{\bf v}\ \) is linear. But \(\ x\mapsto x^2\ \) and \(\ x\mapsto x+1\ \) are not linear. The last is //affine// instead. For a cartesian or vector space any rotation, reflection, stretching or skewing which preserves the origin is a linear map, a linear transformation. 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 producing new functions. Expected value of a random variable is a linear function (but 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. | + | eg \(\ {\bf v}\mapsto c{\bf v}\ \) is linear. But \(\ x\mapsto x^2\ \) and \(\ x\mapsto x+1\ \) are not linear. The last is //affine// instead. For a cartesian or vector space any rotation, reflection, stretching or skewing which preserves the origin is a linear map, a linear transformation. For the appropriate space of [[discussion#functions]] differentiation, definite integral and indefinite integral with a fixed starting point are linear maps, linear operations on those functions producing new functions. Expected value of a random variable is a linear function (but 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 [[discussion#matrix]] algebra and notation to work with transformations including linear and affine ones. | We can use [[discussion#matrix]] algebra and notation to work with transformations including linear and affine ones. | ||
| + | ===linear transformations with matrices=== | ||
| + | Now consider [[applications#square matrices]]. For matrix multiplication with `n`-dimenional square matrices we have an //Identity//, the //Unit// for multiplication, like `1` in the scalars, | ||
| + | \[%%I=\left[\matrix{ | ||
| + | 1&0&\cdots&0\cr | ||
| + | 0&1&\ddots&\vdots\cr | ||
| + | \vdots&\ddots&\ddots&0\cr | ||
| + | 0&\cdots&0&1\cr | ||
| + | }\right]\ \text{ because }\ IA=AI=A=1A.%%\] | ||
| + | Consider: ` X=xI=[ [x,0],[0,x] ],` 2\({\times}\)2 matrix `A` and the 2D vector where `V` is \(\bf v\) written as single column matrix. \(\ xA=XA=AX\ \) and \(x\bf v\) corresponds to \(XV\). | ||
| + | |||
| + | `X` is the simple linear transformation in 2D geometric space of scaling by `x` around the origin. In 2D vector space it is stretching the vectors by `x`, identical to scalar multiplication. Multiplying another 2\({\times}\)2 matrix results in composing the transformations into a single transformation, in the case of `X` it is the same as scalar multiplication by `x`. | ||
| + | |||
| + | Within a **[[#space]]**, given a **[[#basis]]**, each **[[linear#linear map|linear transformation]]** can be represented by a square matrix and every appropriately sized matrix is a linear transformation. Transformations can be thought of as mappings, or as **[[#functions]]**. Matrices can represent these very usefully. In two dimensional space here are some examples: | ||
| + | |||
| + | \(\left[\matrix{3 & 0 \cr 0 & 3 }\right],\quad\) Scale by 3 around origin\\ \\ | ||
| + | \(\left[\matrix{0 & -1 \cr 1 & 0 }\right],\quad\) Rotate right angle anticlockwise around origin\\ \\ | ||
| + | \(\left[\matrix{1 & 0 \cr 0 & -1 }\right],\quad\) Reflect on horizontal axis (negate vertical axis)\\ \\ | ||
| + | \(\left[\matrix{-1 & 0 \cr 0 & -1 }\right],\quad\) Rotate half circle around origin (reflect vert then horiz)\\ \\ | ||
| + | \(\left[\matrix{0 & 0 \cr 0 & 1 }\right],\quad\) Projection onto vertical axis (zero the horiz component)\\ \\ | ||
| + | \(\left[\matrix{1 & \frac12 \cr 0 & 1 }\right],\quad\) Horizontal shear (add half vert component to horiz one)\\ \\ | ||
| + | \(\left[\matrix{3 & 0 \cr 0 & \frac12 }\right],\quad\) Squeeze (asymetrically)\\ \\ | ||
| + | \(\left[\matrix{\cos2\theta & \sin2\theta \cr \sin2\theta & -\cos2\theta}\right],\quad\) Reflection across line through origin, angle `theta` to horizontal | ||
| + | |||
| + | Continued [[discussion#complex-as-matrix|in the main discussion]]. | ||
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