Exercise · ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-005 · ★☆☆ Basic
What is the matrix of the \(90^\circ\) counterclockwise rotation of \(\mathbb{R}^2\)?
Choose one option.
Show answer
- A. \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\)
Explanation
The rotation sends \(e_1=(1,0)\) to \((0,1)\) and \(e_2=(0,1)\) to \((-1,0)\); those are the columns. The matrix \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\) rotates clockwise and \(\operatorname{diag}(1,-1)\) is a reflection.
Review the concept: Matrix of a linear transformation
The matrix \([T]\) of a linear transformation \(T\), with respect to given bases, has as columns the images of the domain basis vectors, written in coordinates of the codomain basis.
\[[T(v)]=[T]\,[v]\]
More practice problems: matrix of a linear transformation
- ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-001 What is the matrix of T(x,y)=(2x+y, 3x-y) in the standard bases?
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- ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-003 The matrix of T is A=[1 2; 0 3]. What is T(2,-1)?
- ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-004 It is known that T(e_1)=(1,4) and T(e_2)=(-2,5). What is the matrix of T in the standard bases?
- ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-006 It is known that T(1,0)=(1,2) and T(0,1)=(3,-1). What is T(2,5)?
- ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-007 S has matrix A=[1 1; 0 1] and T has matrix B=[2 0; 0 3]. What is the matrix of T∘ S?
- ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-008 The derivative D acts on polynomials of degree at most 2. What is its matrix in the basis \1, x, x^…
- ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-009 The matrix of T in the standard bases is [1 2; 3 4] and T(v)=(5,11). What is v?
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