- All lines must remain lines
- Origin must remain fixed in places
- These two properties mean that grid lines stay parallel and evening spaced.
- Only need to record where the basis vectors end in order to model the transformation
- Since each vector began as a linear transformation of the basis vectors, the transformed vectors remain a linear combination of the new, transformed basis vectors $$\begin{bmatrix} a & c\ b & d \end{bmatrix}\begin{bmatrix} x\ y \end{bmatrix} = x \begin{bmatrix} a\ b \end{bmatrix} + y \begin{bmatrix} c\ d \end{bmatrix} = \begin{bmatrix} ax + by\ cx + dy \end{bmatrix}$$
- Let $\begin{bmatrix}
0 & 1\
0 & 1
\end{bmatrix}$ denote a shear matrix and $\begin{bmatrix}
0 & -1\
1 & 0
\end{bmatrix}$ denote a rotation. Then the following is true:
$$\begin{bmatrix}
1 & 1\
0 & 1
\end{bmatrix} \bigg(\begin{bmatrix}
0 & -1\
1 & 0
\end{bmatrix} \begin{bmatrix}
x\
y
\end{bmatrix} \bigg) = \begin{bmatrix}
1 & -1\
1 & 0\
\end{bmatrix}\begin{bmatrix}
x\
y
\end{bmatrix}$$
- These two operations are equivalent to applying the rotation and then the shear.
- Read right to left, think function notation
$f(g(x))$
$AB \ne BC$ $ABC = (AB)C = A(BC)$