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9.1

Question 1

Let $\vec{x}= \left( \begin{smallmatrix} 3 \\ -1 \end{smallmatrix} \right)$ . What is $\vec{x}$ ’s image under $g$ ?

$\begin{equation*} g \colon \, \mathbb{R}^2 \to \mathbb{R}^2 \\ \quad \vec{x} \mapsto \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \vec{x} \end{equation*}$

$\begin{pmatrix} 5 \\ 8 \end{pmatrix}$

$\begin{pmatrix} -2 \\ 1 \end{pmatrix}$

$\begin{pmatrix} 1 \\ 2 \end{pmatrix}$

$\begin{pmatrix} 5 \\ 5 \end{pmatrix}$

Question 2

What is the pre-image of $y=-1$ under $q$ ?

$\begin{equation*} q \colon \, \mathbb{R} \to \mathbb{R} \\ \quad x \mapsto \frac{1}{5}x \end{equation*}$

$-5$

$\{\}$

$\frac{1}{2}$

$\frac{-1}{2}$

Question 3

Let $j$ be a linear transformation. What is the matrix associated to $j$ ?

$\begin{equation*} j \colon \, \mathbb{R}^3 \to \mathbb{R}^3 \\ \quad \begin{pmatrix} x \\ y \\ z \end{pmatrix} \mapsto \begin{pmatrix} 0 \\ x-z \\ x \end{pmatrix} \end{equation*}$

$\begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & -1 \\ 1 & 0 & 0 \end{pmatrix}$

$\begin{pmatrix} 0 \\ 1 -1 \\ 1 \end{pmatrix}$

$\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$

$\begin{pmatrix} 0 & 0 \\ 1 & -1 \\ 1 & 0 \end{pmatrix}$

Question 4

What does this linear transformation do (mouse over the image) ?

Exercise 9.4 : Mont Blanc (credits wikipedia)

reflection

projection

stretching and rotation

rotation

Question 5

What is the matrix associated to this linear transformation (mouse over the image) ?

Exercise 9.5 : Sanshiro pond (credits wikipedia)

$\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

$\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$

$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Question 6

Which function is a linear transformation ?

$c \colon \, \mathbb{R} \to \mathbb{R}$ ,

$x \mapsto \frac{1}{-x}$

$d \colon \, \mathbb{R} \to \mathbb{R}$ ,

$x \mapsto 3^{-x}$

$e \colon \, \mathbb{R} \to \mathbb{R}$ ,

$x \mapsto -1$

$f \colon \, \mathbb{R} \to \mathbb{R}$ ,

$x \mapsto x$