A First Course in Linear Algebra · Problem C61

Vector Spaces — Robert Beezer

Mathematics — Linear Algebra · column-space-descriptions

Problem

Let

A

be the matrix below, and find the indicated sets with the requested properties.

A = \begin{bmatrix} 2 & -1 & 5 & -3 \\ -5 & 3 & -12 & 7 \\ 1 & 1 & 4 & -3 \end{bmatrix}

1. A linearly independent set

S

so that

\mathcal{C}(A) = \langle S \rangle

and

S

is composed of columns of

A

. 2. A linearly independent set

S

so that

\mathcal{C}(A) = \langle S \rangle

and the vectors in

S

have a nice pattern of zeros and ones at the top of the vectors. 3. A linearly independent set

S

so that

\mathcal{C}(A) = \langle S \rangle

and the vectors in

S

have a nice pattern of zeros and ones at the bottom of the vectors. 4. A linearly independent set

S

so that

\mathcal{R}(A) = \langle S \rangle