I am currently self studying linear algebra, using Axler's text. And I always find myself trying to picture every theorem and proposition by considering some low dimensional cases, in order to understand why the theorems make sense. As I get to the chapter for eigenvalues and eigenvectors, things are getting a bit abstract.
Specifically, he proved that every linear operator over an odd dimensional real vector space must have an eigenvalue by using induction, he considered the case where $U$ is a $2$ dimensional invariant subspace of $V$, set $W$ such that $U\oplus W=V$. Then consider the projection of $T\vec w$ onto $W$ along $U$ as an operator in $L(W)$, which must have an eigenvalue $\lambda$ by induction hypothesis. In the end, he showed that $T-\lambda I$ is not injective, thus $\lambda$ is also an eigenvalue of $T$.
I understand everything he's written, but failed to grasp the bigger picture as I can only imagine vector space up to $3$ dimensions (in which case $W$ is only one dimensional). The construction is remarkable for me, as two different linear operators have the same eigenvalue, which I have no idea what the picture would look like (e.g. where does the eigenvector lie on) in higher dimensional spaces.
So my question: Is understanding linear algebra through picture important? In some sense, I feel like it helps me understand more why some theorems would be true, but the proofs/constructions do not rely on picture at all as they mostly involve algebraic manipulations.
All of scientific and mathematical knowledge proceeds from what we directly understand as self-evident to what must logically follow from that. So it is important to understand that that fact about the nature of knowledge applies to any subfield of mathematics.
Easily perceived examples in lower dimensions are an important starting point in grasping the truth of propositions applied by logic to higher dimensions.
Eigenvectors and eigenvalues depend fundamentally upon the concepts of $\textbf{direction}$ and $\textbf{length}$ which we understand by direct perception in lower dimensions.
We can construct lower dimensional examples of linear transformations and directly understand why they leave the directions of certain vectors unchanged, changing perhaps only their length. This is important to understanding the possibility, confirmed by reason, that the same phenomenon can exist in higher dimensions.