Space Projection of singular matrix

Question

Describe the space that $A$ projects onto and the space that $A$ projects along$$A = \begin{bmatrix} 0 & 1 \\0&1 \end{bmatrix}.$$

A matrix, $M$ is a perpendicular projection operator on the column space of $M \iff MM = M $ and $M = M^T.$ Clearly, $A$ does not satisfy this theorem. How do I describe the space that $A$ projects onto and the space that $A$ projects along? I need help.

Answer 1

First you need to check that $A$ projects, that is $A^2=A$. That is true.

Then the subspace it projects to is the range of $A$ (I hope you can find it) and the space it projects along is the nullspace of $A$ (you should be able to find it too).