The rest of the videos from 3Blue1Brown are pretty awesome as well: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
It that all it is, or am I misunderstanding. Everybody seems to think that this field of math is very important, I feel like I should know it by now.
You are correct that the fundamentals (Vectorized computation, aka matrices) are often introduced early on. The level at which you interface with linear algebra often scales with the level of math you are at. You can add them , integrate them , and use them to solve some PDEs . You can see how all these "levels" of linear algebra would be accompanied by another math class pushing you to utilize linear algebra at a higher level than before. It appears that you're at the first level. I've actually never done  in the classroom, but  was introduced to me in a class on differential equations that utilized matrices as a possible solution route.
If you are interested in learning more, Khan academy has good linear algebra videos.
See MIT OCW Scholar's Linear Algebra offering: https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algeb...
Take a look at the final exam and see if it seems like something you've seen. If not, I strongly urge you to consider studying the subject.
A book that shows linear algebra's applications at a more advanced level is Strang's Computational Science and Engineering: http://math.mit.edu/~gs/cse/
I don't know about UK curriculum but the colleges in USA like MIT/Stanford/Princeton/etc have courses focusing on linear algebra (matrix math). Example:
One of those classes is often taken as an elective to meet one of the math requirements for graduation.
(Here I use "calculus" to refer calculus courses without an emphasis on proofs.)
My roommate is studying math and music to be a teacher and i advised him to take calc first and then linear algebra and until now it's working out.
Source: Currently studying CS Bsc. in Karlsruhe where we are required to take Calc 1 and 2 and Linear Algebra 1 and 2, but in parallel (so first Calc 1 and LA 1 and then Calc 2 and LA 2, it's also doable).
A lot of theorems like SVD and eigenvectors seem like they'd carry over, but they are always proven over complex numbers.
For example, see https://youtu.be/c194lY79jOM minute 16:30
The course makes no specifics for finite fields but does illustrate that linear algebra works for any field, finite or not.