structures:matrix:linear
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structures:matrix:linear [2025/03/01 14:02] simon |
structures:matrix:linear [2025/03/03 11:16] (current) simon |
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| 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 \(XV\). | ||
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| + | `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: | ||
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| + | \(\left[\matrix{3 & 0 \cr 0 & 3 }\right],\quad\) 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]]. | ||
structures/matrix/linear.1740798122.txt.gz ยท Last modified: 2025/03/01 14:02 by simon
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