Problem on matrices as transformations
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Let $A= \begin{pmatrix}0 & -1 \\ -2 & 0 \end{pmatrix}$.
Sketch how the following shape gets transformed by application of $A$:
Solution
Recall that
Matrices as transformations
To graphically explore the application of a matrix, see how it acts on the standard vectors $(1,0)$ and $(0,1)$. Then piece together the effect on the overall shape by considering the $x$ and $y$ behaviors separately.
Matrix multiplication
We begin by seeing how $A$ transforms the vector $(1,0)$ by performing the matrix multiplication: $$ \begin{pmatrix}0 & -1 \\ -2 & 0 \end{pmatrix} \begin{pmatrix} 1\\0 \end{pmatrix} = \begin{pmatrix}0 \\ -2 \end{pmatrix}$$
We see that $(1,0)$ gets transformed to $(0, -2)$.
Matrix multiplication
Similarly, $A$ transforms the vector $(0,1)$ into $$ \begin{pmatrix}0 & -1 \\ -2 & 0 \end{pmatrix} \begin{pmatrix} 0\\1 \end{pmatrix} = \begin{pmatrix}-1 \\ 0 \end{pmatrix}$$
We see that $(0,1)$ gets transformed to to $(-1,0)$.
Graphically:
We see that $A$ acts like the combination of rotating 90 degrees clockwise, mirroring about the $y$ axis, and then doubling the $y$ component.
Performing the same operations on the provided shape, we see the transformation:
We conclude that the transformed shape is:
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