Question Bank • QB_EM_LA_006 MCQ • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Linear and orthogonal transformations

Question Statement:

The linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ that reflects every vector about the line $y = x$ has the standard matrix representation:

Options:

  • (A) $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
  • (B) $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$
  • (C) $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
  • (D) $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

Answer Key:

A

Step-by-Step Solution & Explanation

Under reflection across the line $y = x$: 1. The basis vector $e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ transforms to $T(e_1) = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. 2. The basis vector $e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ transforms to $T(e_2) = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$. The standard matrix is formed by these column vectors: $[T] = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ Notice that $[T]^T [T] = I$ and $\det([T]) = -1$, which confirms it is an orthogonal reflection.
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