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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![enter image description here]1

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$E_2$ 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?

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    $\begingroup$ It's the last part that I'm unsure of. This is supposed to be easy but I'm having a mental block now.. I've row reduced just like him. From this I get two equations v1 = 0, v2 = 0. So? Do I let v3 = t (free variable) and then all vectors in the kernel are of the form t<0,0,1>, therefore E2 equals span(<0,0,1>). IS that reasoning correct? $\endgroup$ – Shad0wNinja Aug 20 '15 at 2:04
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    $\begingroup$ @Shad0wNinja Yep, that's the reasoning! $\endgroup$ – pjs36 Aug 20 '15 at 2:43

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