structures:matrix:linear
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| 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: | 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: | ||
| - | \(\left[\matrix{0 & 0 \cr 0 & 1 }\right],\quad\) Projection onto vertical axis\\ \\ | + | \(\left[\matrix{3 & 0 \cr 0 & 3 }\right],\quad\) Scale by 3 around origin\\ \\ |
| - | \(\left[\matrix{1 & 2 \cr 0 & 1 }\right],\quad\) Horizontal shear\\ \\ | + | \(\left[\matrix{0 & -1 \cr 1 & 0 }\right],\quad\) 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)\\ \\ | \(\left[\matrix{3 & 0 \cr 0 & \frac12 }\right],\quad\) Squeeze (asymetrically)\\ \\ | ||
| \(\left[\matrix{\cos2\theta & \sin2\theta \cr \sin2\theta & -\cos2\theta}\right],\quad\) Reflection across line through origin, angle `theta` to horizontal | \(\left[\matrix{\cos2\theta & \sin2\theta \cr \sin2\theta & -\cos2\theta}\right],\quad\) Reflection across line through origin, angle `theta` to horizontal | ||
| Continued [[discussion#complex-as-matrix|in the main discussion]]. | Continued [[discussion#complex-as-matrix|in the main discussion]]. | ||
structures/matrix/linear.1740899653.txt.gz · Last modified: 2025/03/02 18:14 by simon
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