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Question

Find a matrix that performs the stated composition in homogeneous coordinates. The translation of R2R^{2} by (1, 1), followed by the scaling transformation (x,y)(2x,7y)(x, y) \rightarrow(2 x, 7 y).

Solution

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Answered 2 years ago
Answered 2 years ago
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The standard matrix for a translation of (1,1)(1,1) in homogeneous coordinates is

[T1]=[101011001][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

[T2]=[200070001][T_2] = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Therefore,

[T2][T1]=[200070001][101011001]=[202077001]\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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