Find Eigenspace/Eigenvalue for a linear transformation

I am currently taking a basic 100 level linear algebra class, and we just went over how to find eigenspace and eigenvalue for a matrix. But on our homework, a question says

Suppose that λ is an eigenvalue of T : V → V, show blah blah

Since a linear transformation can be written as a matrix, can I just think of it as solving for eigenvalue/space of a matrix? Is there anything fundamentally different? Thank you!

• Well, whether or not you can (or should) think of $T$ as a matrix depends on what the "blah blah" is – Omnomnomnom Dec 2 '16 at 2:23
• @Omnomnomnom "Show that the corresponding eigenspace is a T-invariant subspace of V." --- I know to show it's a subspace, I need to show T is not an empty set, and show addition and scale multiplication still in T. But I am not sure what it means by an eigenvalue of a linear transformation. We only talk about matrices with eigenvalue/space. – Vanya Dec 2 '16 at 2:29
• You mean that you need to show that V is a subspace (that it's closed under addition and multiplication by scalars). You also need to show that it's T-invariant, which is something you should look up in your notes or textbook. "$V$ is $T$-invariant" means that for any $v \in V$, $T(v)$ is also in $V$. – Omnomnomnom Dec 2 '16 at 2:40
• This is a question where anything you might do to write $T$ as an explicit matrix or "compute the eigenvalues/eigenspaces" is besides the point. A question like this comes down purely to understanding the definitions at play here, and how they fit together. – Omnomnomnom Dec 2 '16 at 2:42
• @Omnomnomnom Thank you! I will try to understand it. – Vanya Dec 2 '16 at 2:45