Exercise 85Linear algebra › Linear mapsHigh schoolAdded 3 Nov 2026

Matrix of a linear map

#matrix representation#canonical basis#linear map

Statement

Let $f:\mathbb{R}^3\to\mathbb{R}^2$ be defined by $f(x,y,z)=(x+2y,\ 3y-z)$ . Write the matrix of $f$ in the canonical bases.

Hint

The columns of the matrix are the images of the canonical basis vectors.

Show solutionHide solution

Images.
$f(1,0,0)=(1,0), \qquad f(0,1,0)=(2,3), \qquad f(0,0,1)=(0,-1).$

Matrix. These are the columns:
$A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 3 & -1 \end{pmatrix}.$

Check. $A\,(x,y,z)^T = (x+2y,\ 3y-z)^T$ , which is $f(x,y,z)$ .