Exercise · ZW-MATRIX-OF-A-LINEAR-TRANSFORMATION-008 · ★★☆ Intermediate
The derivative \(D\) acts on polynomials of degree at most 2. What is its matrix in the basis \(\{1,\,x,\,x^2\}\)?
Choose one option.
A \(\begin{pmatrix}0&0&0\\1&0&0\\0&2&0\end{pmatrix}\) B \(\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}\) C \(\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}\) D \(\begin{pmatrix}0&1&0\\0&0&2\\0&0&0\end{pmatrix}\) Show answer D. \(\begin{pmatrix}0&1&0\\0&0&2\\0&0&0\end{pmatrix}\)← Back to the question
Explanation \(D(1)=0\), \(D(x)=1\) and \(D(x^2)=2x\). In coordinates: \((0,0,0)\), \((1,0,0)\) and \((0,2,0)\), which are the three columns.
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.
\[[T(v)]=[T]\,[v]\]
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