Matrix Composition: Reflection and Dilation

00:00 6 marks

Consider a transformation in the coordinate plane that involves two sequential steps: first, reflect a generic point \((x,y)\) about the line \(y=x\), and then enlarge the reflected point by a scale factor of 2 centered at the origin. The overall transformation is a composition of the reflection and dilation transformations.

Part a

5 minutes 3 marks

Question

Determine the matrix that represents a reflection about the line \(y=x\). Apply this reflection matrix to a generic point \((x,y)\) and express the coordinates of the reflected point.

Hint Hint

Hint

Recall the standard reflection matrix for the line \(y=x\). To apply the transformation, multiply the reflection matrix by the column vector representing the point \((x,y)\).

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Complete Answer

The reflection matrix about the line \(y=x\) is \(R = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\). Applying this matrix to a point \((x,y)\) gives \(\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix} \begin{pmatrix}x\\ y\end{pmatrix} = \begin{pmatrix}y\\ x\end{pmatrix}\). Thus, the reflected point is \((y,x)\).

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Part b

5 minutes 3 marks

Question

A dilation with a scale factor of 2, centered at the origin, is applied after the reflection from part (a). Determine the matrix for this dilation. Then, find the matrix that represents the composition of the reflection followed by the dilation. Finally, state the coordinates of the point obtained by applying this composite transformation to a generic point \((x,y)\).

Hint Hint

Hint

The dilation matrix for a scale factor of 2 is a diagonal matrix. To find the composite transformation matrix, multiply the dilation matrix by the reflection matrix (in the correct order). Apply the composite matrix to \((x,y)\) to find the final transformed point.

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The dilation matrix with a scale factor of 2 is \(D = \begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}\). The composite transformation matrix is obtained by multiplying the dilation matrix by the reflection matrix: \(C = D \cdot R = \begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix} \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix} = \begin{pmatrix}0 & 2\\ 2 & 0\end{pmatrix}\). Applying this composite matrix to a point \((x,y)\) gives \(\begin{pmatrix}0 & 2\\ 2 & 0\end{pmatrix} \begin{pmatrix}x\\ y\end{pmatrix} = \begin{pmatrix}2y\\ 2x\end{pmatrix}\). Thus, the final transformed point is \((2y,2x)\).

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