Exercises: Work with 2x2 Matrices as Transformations

To understand what a $2 \times 2$ matrix $A$ does to the entire plane, you need to track only two special vectors. Which two?

The unit square has vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Under any $2 \times 2$ transformation $A$, where does the origin $(0,0)$ always go?

The $2 \times 2$ identity matrix $I$ is applied to the unit square. What is the area of the image?

For $A = \begin{bmatrix}3 & 1\\0 & 2\end{bmatrix}$, where does $\mathbf{e}_2 = \begin{bmatrix}0\\1\end{bmatrix}$ map under $A$?

Matrix $A = \begin{bmatrix}2 & 0\\0 & 3\end{bmatrix}$. Where does the vertex $(1,1)$ of the unit square map?

The unit square has area 1. Matrix $A = \begin{bmatrix}3 & 1\\0 & 2\end{bmatrix}$ is applied. What is the area of the image parallelogram?

A region of area 4 is transformed by matrix $A$ with $\det(A) = -3$. What is the area of the image?