I'm studying for an exam and I don't understand how my prof finds the basis for eigenspaces using the matrix representation of a linear map. Once I find an eigevalue then how do I find the basis for its eigenspace. I've attached a screenshot of the part that I don't understand (from an example). Can someone please explain it to me in detail? Thanks]1
$E_2$ means the eigenspace of $\lambda=2$. The eigenspace of an eigenvalue (2 in this case) tells us all the possible eigenvectors associated to that eigenvalue. The kernel of a matrix $A$ is an element (vector in this case) $x$ such that $Ax=0$. Now from line 1 through 4, he is just row reducing the matrix. I'm assuming you know that from early in the course. To find a vector $x \in ker(A-2I)$ (the eigenspace of 2), we can write out the matrix as 3 equations and solve the homogeneous equations $Ax=0$. Can you write the matrix as equations and finish from there?