Exercise 1218Linear algebra › Vector spaces and subspacesL1Added 3 Nov 2026

Column space and row space

#column space#row space#rank

Statement

Let
$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix}.$
Describe the column space and row space of $A$ .

Hint

All rows are proportional. The column space is spanned by the columns.

Show solutionHide solution

Row space. All rows are multiples of $(1,2,3)$ , so the row space is $\operatorname{span}((1,2,3))$ , dimension $1$ .

Column space. All columns are multiples of $(1,2,3)^T$ , so the column space is $\operatorname{span}((1,2,3)^T)$ , dimension $1$ .

Rank. $\operatorname{rank}(A)=1$ .

Kernel. $\dim\ker A=3-1=2$ . Solve $x+2y+3z=0$ : basis $(-2,1,0),(-3,0,1)$ .

Conclusion. Row and column spaces are both lines, and $\ker A$ is a plane.