A First Course in Linear Algebra · Problem CRS.C20

Matrices — Robert Beezer

Mathematics — Linear Algebra · column-space-row-space

Problem

For each matrix below, find a set of linearly independent vectors

X

so that

\langle X \rangle

equals the column space of the matrix, and a set of linearly independent vectors

Y

so that

\langle Y \rangle

equals the row space of the matrix.

A = \begin{bmatrix} 1 & 2 & 3 & 1 \\ 0 & 1 & 1 & 2 \\ 1 & -1 & 2 & 3 \\ 1 & 1 & 2 & -1 \end{bmatrix} \quad B = \begin{bmatrix} 1 & 2 & 1 & 1 & 1 \\ 3 & 2 & -1 & 4 & 5 \\ 0 & 1 & 1 & 1 & 2 \end{bmatrix} \quad C = \begin{bmatrix} 2 & 1 & 0 \\ 3 & 0 & 3 \\ 1 & 2 & -3 \\ 1 & 1 & -1 \\ 1 & 1 & -1 \end{bmatrix}

From your results for these three matrices, can you formulate a conjecture about the sets

X

and

Y