Exercise 1920Linear algebra › Vector spaces and subspacesL1Added 3 Nov 2026

Matrix of a linear map in a given basis

#matrix representation#basis#linear map

Statement

Let $f:\mathbb{R}_2[X]\to\mathbb{R}_2[X]$ be defined by $f(P)=P'$ . Write the matrix of $f$ in the canonical basis $(1,X,X^2)$ .

Hint

Compute the image of each basis vector and express it in the basis.

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Images. $f(1)=0$ , $f(X)=1$ , $f(X^2)=2X$ .

Coordinates.
$f(1)=(0,0,0), \quad f(X)=(1,0,0), \quad f(X^2)=(0,2,0).$

Matrix. The columns are the coordinate vectors:
$M = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{pmatrix}.$

Conclusion. The matrix of $f$ in the canonical basis is $M$ . It is nilpotent of index $3$ : $M^3=0$ .