Find the column space of the matrix $ \left(\begin{smallmatrix} 1 & 0 & -1 & 0 & 1\\ 0 & 1 & -1 & 2 &0\\ \end{smallmatrix}\right) $

Question

I have the following matrix

$$ \begin{pmatrix} 1 & 0 & -1 & 0 & 1\\ 0 & 1 & -1 & 2 &0\\ \end{pmatrix} $$

And I am unsure as to how to write the column space/image for the transformation it represents.

I know that, for a Mapping $M:S\to R$, $\operatorname{Im}(M)$ is the subspace of elements $x$ of $R$ for which there is some element $y$ of $S$ such that $M(y)=x$.

Should I write the Image as $$\operatorname{Im}(M)= \left(\begin{matrix} x_{1}-x_3+x_5\\ x_{2}-x_3+2x_4\\ \end{matrix}\right) $$

for $x_1,x_2,x_3,x_4,x_5 \in\mathbb R$?????

Answer 1

That is not the image space, that is just another way to write the transformation. The image space is the entire $R^2$, since for any vector $(x,y)$, at least the vector $(x,y,0,0,0)$ is mapped onto it.

Answer 2

The image space is generated by the images of the vectors in the basis of $\mathbf R^5$ chosen to represent the transformation. Now, these images are precisely the $5$ column vectors. As they contain the vectors $\;\begin{pmatrix} 1\\0\end{pmatrix}$ and $\;\begin{pmatrix}0\\1\end{pmatrix}$ , which are a basis of $\mathbf R^2$, this means the image space is equal to $\mathbf R^2$.