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Linear Algebra

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Are These Transformations Linear?

LINALG-N1M1I1

Each of the following choices describes a function $\mathbb{R}^3\to\mathbb{R}^2$ and is written in the form:

$$f(x)=\begin{pmatrix}y_1\\\ y_2\end{pmatrix}$$

...where:

$$x=\begin{pmatrix}x_1\\\ x_2\\\ x_3\end{pmatrix}\in\mathbb{R}^3$$

Which of these choices are linear transformations?

Select ALL that apply.

A

$f(x)=\begin{pmatrix}x_1x_2\\\ x_1\end{pmatrix}$

B

$f(x)=\begin{pmatrix}x_1+x_2\\\ 0\end{pmatrix}$

C

$f(x)=\begin{pmatrix}x_1^2\\\ x_2\end{pmatrix}$

D

$f(x)=\begin{pmatrix}x_1\\\ 3^2x_2\end{pmatrix}$

E

$f(x)=\begin{pmatrix}1\\\ 1\end{pmatrix}$