AP Precalculus / Unit 4: Functions Involving Parameters, Vectors, and Matrices
4.12 Linear Transformations and Matrices
Notes
A $2\times2$ matrix maps vectors in the plane: $\vec v\mapsto A\vec v$.
- Rotation by $\theta$: $\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$.
- The columns show where $\langle1,0\rangle$ and $\langle0,1\rangle$ go.
Practice
$\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ rotates vectors by:
$\displaystyle \theta=\frac{\pi}{2}$.
$\begin{bmatrix}2&0\\0&2\end{bmatrix}$ does what?
Every vector doubles.
The first column of $A$ is where $A$ sends:
$A\langle1,0\rangle$ is the first column.
Ready for more? Take the Unit 4 test.