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Suppose $A \in M_{2,3}(\mathbb{R})$ is a matrix and

$A \mathbf{e_1} = {\left[\begin{array}{c} 3\cr -2 \end{array}\right]}$ , $A \mathbf{e_2} = {\left[\begin{array}{c} 3\cr -1 \end{array}\right]}$ , and $A \mathbf{e_3} = {\left[\begin{array}{c} 3\cr 0 \end{array}\right]}$ .

What is $A( 4 \mathbf{e_1} - 4 \mathbf{e_2} - 4 \mathbf{e_3})$ ? Enter your answer as a coordinate vector of the form <1,2>.
$A( 4 \mathbf{e_1} - 4 \mathbf{e_2} - 4 \mathbf{e_3}) =$

Find the matrix for the linear transformation $f$ (relative to the standard basis in the domain and codomain). That is, find the matrix $A$ such that $f(\mathbf{x}) = A \mathbf{x}$ . For instance, enter [ [1,2], [3,4] ] for the matrix $\left[\begin{array}{cc} 1 &2\cr 3 &4 \end{array}\right]$ .
$A=$

Find a formula for the linear transformation $f$ . Enter your answer as a coordinate vector with the variables $x$ , $y$ , and $z$ in its components.
$f(x,y,z) =$

Find bases (i.e., maximal independent sets) for the kernel and image of $f$ . Please enter your answers as comma separated lists of vectors of the form <1,2> or <1,2,3>, as appropriate.
$\mathrm{Kernel}(f) = \mathrm{span} \Big\lbrace$ $\Big\rbrace.$
$\mathrm{Image}(f) = \mathrm{span} \Big\lbrace$ $\Big\rbrace.$

The linear transformation $f$ is (select all that apply):
A. bijective (an isomorphism)
B. surjective (onto)
C. injective (one-to-one)
D. none of these

You can earn partial credit on this problem.