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Exercise 10.4.4

Avg difficulty 3.0

Let $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a linear transformation defined by

T\left(\left[\begin{array}{c}a\\ b\end{array}\right]\right)=\left[\begin{array}{c}a+b\\ a-b\end{array}\right].

Consider the two bases

B_{1}=\left\{\left[\begin{array}{r}1\\ 0\end{array}\right],\left[\begin{array}{r}-1\\ 1\end{array}\right]\right\}

and

B_{2}=\left\{\left[\begin{array}{r}1\\ 1\end{array}\right],\left[\begin{array}{r}1\\ -1\end{array}\right]\right\}.

Find the matrix $M_{B_{2},B_{1}}$ of $T$ with respect to the bases $B_{1}$ and $B_{2}$ .

Selinger, Matrix Theory and Linear Algebra, Linear transformation of vector spaces, The matrix of a linear transformation, Exercise 10.4.4

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.