Question

Determine if the columns of the a span r2. a = 6 −12 −2 4

Answer 1

If


`\mathbf A=\begin{bmatrix}6&-12\\-2&4\end{bmatrix}`

then notice that the columns satisfy


`\begin{bmatrix}6\\-2\end{bmatrix}=-\frac12\begin{bmatrix}-12\\4\end{bmatrix}\implies\begin{bmatrix}6\\-2\end{bmatrix}+\frac12\begin{bmatrix}-12\\4\end{bmatrix}=\mathbf 0`

which means the columns are linearly dependent and thus only span a subspace of
`\mathbb R^2`.

Whether you actually meant to write


`\mathbf A=\begin{bmatrix}6&-2\\-12&4\end{bmatrix}`

would not alter the answer - the columns do not span
`\mathbb R^2` - but this time


`\begin{bmatrix}6\\-12\end{bmatrix}=-\frac13\begin{bmatrix}-2\\4\end{bmatrix}`