Consider this matrix transformation: $\newline$ $\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right]$ $\newline$ What is the geometric effect of this transformation? $\newline$ Choose $1$ answer: $\newline$ (A) A reflection across the $y$ axis $\newline$ (B) A reflection across the line $y=x$ $\newline$ (C) A rotation about the origin by $90^{\circ}$ $\newline$ (D) A rotation about the origin by $180^{\circ}$

Full solution

Q. Consider this matrix transformation:

\newline

\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right]

\newline

What is the geometric effect of this transformation?

\newline

Choose

1

answer:

\newline

(A) A reflection across the

y

axis

\newline

(B) A reflection across the line

y=x

\newline

90^{\circ}

\newline

(D) A rotation about the origin by

180^{\circ}

Identify transformation: Identify the matrix transformation. $\newline$ The matrix given is: $\newline$ $\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}$ $\newline$ This is a $2 \times 2$ matrix that can be applied to vectors in the plane.
Effect on basis vectors: Understand the effect of the matrix on basis vectors. To understand the transformation, we can look at what happens to the standard basis vectors $i = [1, 0]$ and $j = [0, 1]$ . Applying the matrix to $i$ : $\begin{pmatrix}0 & -1 \ 1 & 0\end{pmatrix} \begin{pmatrix}1 \ 0\end{pmatrix} = \begin{pmatrix}0 \ 1\end{pmatrix}$ The vector $i$ is rotated $90$ degrees counterclockwise to become $j$ .
Apply to basis vectors: Apply the matrix to the second basis vector $j$ : $\begin{pmatrix}0 & -1 \ 1 & 0\end{pmatrix} \begin{pmatrix}0 \ 1\end{pmatrix} = \begin{pmatrix}-1 \ 0\end{pmatrix}$ The vector $j$ is rotated $90$ degrees counterclockwise to become $-i$ .
Conclude geometric effect: Conclude the geometric effect of the transformation. Both basis vectors are rotated $90$ degrees counterclockwise. Therefore, the matrix represents a rotation by $90$ degrees counterclockwise about the origin.