Back to Exercise: Multiply matrices and vectors

Exercises: Multiply Matrices and Vectors

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-11

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Recall / Warm-Up

How is the vector $\mathbf{v} = \langle 3, -5 \rangle$ written as a column matrix?

$\begin{bmatrix}3 & -5\end{bmatrix}$

$\begin{bmatrix}3\\-5\end{bmatrix}$

$\begin{bmatrix}-5\\3\end{bmatrix}$

$\begin{bmatrix}3 & 0\\0 & -5\end{bmatrix}$

To find the first entry of $A\mathbf{v}$ where $A = \begin{bmatrix}a&b\\c&d\end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix}v_1\\v_2\end{bmatrix}$ , which computation is correct?

$a \cdot v_1$ only (just the top-left entry of $A$ times the top entry of $\mathbf{v}$ )

$a \cdot v_1 + b \cdot v_2$ (row 1 of $A$ dotted with $\mathbf{v}$ )

$c \cdot v_1 + d \cdot v_2$ (row 2 of $A$ dotted with $\mathbf{v}$ )

$a \cdot v_1 \cdot b \cdot v_2$ (product of all relevant entries)

The matrix $A = \begin{bmatrix}k&0\\0&k\end{bmatrix}$ maps every vector to a new vector. What geometric transformation does $A$ represent?

A rotation by $k$ degrees

A reflection over the $x$ -axis

A uniform scaling by factor $k$

A shear along the $x$ -axis

Fluency Practice

Which dimensions describe the product $A\mathbf{v}$ when $A$ is $2 \times 2$ and $\mathbf{v}$ is a $2 \times 1$ column vector?

$4 \times 1$

$2 \times 2$

$2 \times 1$

$1 \times 2$

Compute $\begin{bmatrix}2 & 1\\3 & 4\end{bmatrix}\begin{bmatrix}1\\2\end{bmatrix}$ . What is the first entry of the result?

Compute $\begin{bmatrix}2 & 1\\3 & 4\end{bmatrix}\begin{bmatrix}1\\2\end{bmatrix}$ . What is the second entry of the result?

Apply the matrix $\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}$ to $\mathbf{v} = \begin{bmatrix}3\\4\end{bmatrix}$ . What is the result?

$\begin{bmatrix}4\\3\end{bmatrix}$

$\begin{bmatrix}-4\\3\end{bmatrix}$

$\begin{bmatrix}3\\-4\end{bmatrix}$

$\begin{bmatrix}-3\\4\end{bmatrix}$

Transformation $R$ (rotation) is applied first, then transformation $S$ (scaling). Which matrix product represents this composition?

$RS\mathbf{v}$

$SR\mathbf{v}$

$R + S$

$S + R$

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