Back to Exercise: Work with 2x2 matrices as transformations

Exercises: Work with 2x2 Matrices as Transformations

Work through each section in order. Show your work where indicated.

Grade 9·20 problems·~30 min·Common Core Math - HS Number and Quantity·standard·hsn-vm-c-12

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Recall / Warm-Up

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 first and last rows of $A$

The standard basis vectors $\mathbf{e}_1 = \begin{bmatrix}1\\0\end{bmatrix}$ and $\mathbf{e}_2 = \begin{bmatrix}0\\1\end{bmatrix}$

Any two perpendicular vectors

The zero vector and the vector of all ones

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?

To $(1,1)$

Depends on the matrix $A$

To $(0,0)$ — the origin always maps to itself

To the center of the image parallelogram

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

Fluency Practice

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

$\begin{bmatrix}3\\0\end{bmatrix}$ (column 1 of $A$ )

$\begin{bmatrix}1\\2\end{bmatrix}$ (column 2 of $A$ )

$\begin{bmatrix}0\\1\end{bmatrix}$ (unchanged)

$\begin{bmatrix}0\\2\end{bmatrix}$

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

$(2, 3)$

$(1, 1)$

$(5, 5)$

$(2, 0)$

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?

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