Exercise · ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-010 · ★★★ Advanced

Let \(T(x,y)=(y,x)\) and the basis \(B=\{(1,1),\,(1,-1)\}\) used in domain and codomain. What is the matrix of \(T\) in \(B\)?

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  • C. \(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\)

Explanation

\(T(1,1)=(1,1)\), whose coordinates in \(B\) are \((1,0)\); \(T(1,-1)=(-1,1)=-(1,-1)\), with coordinates \((0,-1)\). These are the columns of the diagonal matrix \(\operatorname{diag}(1,-1)\). The standard matrix is \(\begin{pmatrix}0&1\\1&0\end{pmatrix}\).

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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

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