Consider two 2×2 matrices representing linear transformations: \(A = \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 4 \\ -2 & 0 \end{pmatrix}\).
(a) Write down the matrix product \(C = A \times B\).
(b) Interpret the composite transformation represented by \(C\) by explaining the effect on a general vector in \(\mathbb{R}^2\).
Question
Setup the matrix multiplication of \(A\) and \(B\) to compute \(C = A \times B\). Show your calculation steps.
Hint
Hint
Hint
Multiply corresponding elements of the rows of \(A\) by the columns of \(B\). Ensure you show each step of your calculation clearly.
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Complete Answer
\(C = \begin{pmatrix} 4 & 8 \\ -6 & 0 \end{pmatrix}\)
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Question
Interpret the composite transformation represented by matrix \(C\). What effect does \(C\) have on a general vector in \(\mathbb{R}^2\)? Provide a brief explanation in terms of scaling, rotation, or shear, and describe how the components of a general vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) are transformed.
Hint
Hint
Hint
Consider the transformation \(C\begin{pmatrix} x \\ y \end{pmatrix}\) and analyze the resulting vector components in terms of \(x\) and \(y\).
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Complete Answer
The composite transformation \(C\) represents a combination of scaling and shearing. When \(C\) is applied to a general vector \(\begin{pmatrix} x \\ y \end{pmatrix}\), the transformed vector is \(\begin{pmatrix} 4x + 8y \\ -6x \end{pmatrix}\). This indicates that the x-component is scaled and sheared as it depends on both \(x\) and \(y\) (\(4x + 8y\)). The y-component is scaled, reflected (due to the negative sign), and sheared as it depends only on \(x\) (\(-6x\)). There is no pure rotation component evident from the form of \(C\) directly, although combinations of shears and scalings can sometimes produce effects similar to rotation in specific contexts.
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