Understanding Linear Algebra Geometrically - Reference Request
I know geometry and I know linear algebra but when I understand a linear algebraic concept geometrically, my head just explodes and things just become so much clearer and easier to understand...not to mention easier to remember or figure out its properties and explain them to others.
Here are a few examples.
Orthogonal matrices - If you think of an orthogonal matrix as a rotation then some of its properties are obvious. Orthogonal matrices are always invertible because rotations can simply be reversed. They always preserve the Euclidean norm because rotating a vector doesn't change its length. Orthogonal matrices forming a group is also easy to see because it is easy to see them satisfying the group axioms.
Determinant - The determinant of a linear transformation can be understood as follows. Start with the (chosen) basis of your domain. It forms a parallelepiped. Call it $P$. It has a certain volume $V(P)$. Now apply your linear transformation $T$ to the chosen basis. A new parallelepiped $T(P)$ is formed and its volume in the range space (embedded in the codomain) is now $V(T(P))$. The determinant (in absolute value) is the ratio of the new volume to the old one. This intuitively explains, for example, why the determinant is zero for non-invertible transformations. The dimension of such a transformation will always be strictly less than the dimension of the domain/codomain so the volume of the transformed parallelepiped will always be zero. I always imagine a parallelepiped in $\mathbb{R}^3$ collapsing onto a plane. This also explains why the determinant of an orthogonal matrix is always $\pm1$ because rotating a parallelepiped won't change its volume. In addition, it kind of helps with the Jacobian determinant and why is the Jacobian "necessary" when transforming variables.
Singular value decomposition - Every matrix having an SVD says the fantastical fact that any linear transformation can be considered a rotation, then a dilation (different directions by different factors), and then a rotation again.
Projection matrices - Imagine an arbitrary vector's shadow onto a line or a plane. I imagine a vector collapsing onto its shadow and properties like $P^2=P$ are immediate for any projector operator $P$. Take this and run with it.
My question is, can anyone point to some good reading material where a geometric interpretation of various linear algebra concepts is offered?
This could be anyone's class/teaching notes, published papers, something from recreational mathematics, or just a good book.
It might be slightly off-topic since you are searching for reading material, but the 3blue1brown YouTube channel is dedicated to this sort of geometric intuitions. More specifically, it presents abstract concept with geometric animations, spanning from neural networks to Fourier transform.
Although it is not restricted to such matter, it has an excellent playlist called Essence of linear algebra which adresses what you seek.
I'm the same way, you pretty much described my experience with Linear Algebra suddenly 'clicking' when it's explained geometrically. I found a really good book that treats alot of different LA concepts in this way, it's Numerical Linear Algebra by Trefethen.
Check out chapter 4 for a geometric breakdown of Singular Value Decomposition!