Matrices as Transformations

A matrix is a machine that moves the plane

A $2\times2$ matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ acts on a vector by $A\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$. The whole transformation is determined by two facts: the first column is the image of $\langle1,0\rangle$, the second column is the image of $\langle0,1\rangle$.

Composition is multiplication — applying $A$ then $B$ is the single matrix $BA$ (rightmost acts first), and in general $AB\neq BA$: order matters.

The determinant measures area: $\det A=ad-bc$, and any region of area $S$ maps to a region of area $|\det A|\cdot S$. If $\det A=0$ the plane collapses onto a line and the transformation cannot be undone — no inverse matrix exists.

Named transformations: rotation by $\theta$ is $\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$ (determinant $1$: areas preserved); dilation by $k$ is $kI$ (determinant $k^2$).

Transition-state models: when a matrix's columns are probability distributions, repeated multiplication $\vec s_{n+1}=T\vec s_n$ pushes a population between states; many systems settle into a steady state $\vec s$ with $T\vec s=\vec s$.