(Solution) - Use Theorem 4 3 3 to find the standard matrix for T -(2025 Original AI-Free Solution)

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Use Theorem 4.3.3 to find the standard matrix for T: R2 ? R2 from the images of the standard basis vectors.
(a) T: R2 ? R2 projects a vector orthogonally onto the x-axis and then reflects that vector about the y-axis.
(b) T: R2 ? R2 reflects a vector about the line y = x and then reflects that vector about the x-axis.
(c) T: R2 ? R2 dilates a vector by a factor of 3, then reflects that vector about the line y = x, and then projects that vector orthogonally onto the y-axis.