Problem on matrices as transformations
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Let $A= \begin{pmatrix}1/2 & 0 \\ 0 & 2 \end{pmatrix}$.
How does the following shape get 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: $$ \begin{pmatrix}1/2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 1\\0 \end{pmatrix} = \begin{pmatrix}1/2 \\ 0 \end{pmatrix}$$
We see $(1,0)$ gets shunk by half to $(1/2, 0)$.
Matrix multiplication
Similarly, $A$ transforms the vector $(0,1)$ into $$ \begin{pmatrix}1/2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 0\\1 \end{pmatrix} = \begin{pmatrix}0 \\ 2 \end{pmatrix}$$
We see $(0,1)$ gets scaled by two to $(0,2)$.
Graphically:
We see that the $x$ component gets cut in half, while the $y$ component is doubled. Hence the shape is transformed to:
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