A First Course in Linear Algebra · Problem CRS.C30

Matrices — Robert Beezer

Mathematics — Linear Algebra · column-space-linear-combinations

Problem

Example CSOCD expresses the column space of the coefficient matrix from Archetype D (call the matrix

A

here) as the span of the first two columns of

A

. In Example CSMCS we determined that the vector

\mathbf{c} = \begin{bmatrix} 2 \\ 3 \\ 2 \end{bmatrix}

was not in the column space of

A

and that the vector

\mathbf{b} = \begin{bmatrix} 8 \\ -12 \\ 4 \end{bmatrix}

was in the column space of

A

. Attempt to write

\mathbf{c}

and

\mathbf{b}

as linear combinations of the two vectors in the span construction for the column space in Example CSOCD and record your observations.