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Question

# Find a matrix that performs the stated composition in homogeneous coordinates. The translation of $R^{2}$ by (1, 1), followed by the scaling transformation $(x, y) \rightarrow(2 x, 7 y)$.

Solution

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Step 1
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The standard matrix for a translation of $(1,1)$ in homogeneous coordinates is

$[T_1] = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$

For the given scaling transformation, the standard matrix in homogeneous coordinates would be

$[T_2] = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Therefore,

\begin{align*} [T_2][T_1] &= \begin{bmatrix} 2 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} 2 & 0 & 2 \\ 0 & 7 & 7 \\ 0 & 0 & 1 \end{bmatrix} \end{align*}

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