I would like to offer my views on the subject and its available resources:
What is it about?
Upto school level mathematics, we usually deal with numbers, like ##x## in all of our equations are just the numbers; following commutativity, associativity and other usual things. For a short course we deal with functions when we study Calculus, but still functions represent numbers only.
When we come to Linear Algebra, it deals with all possible mathematical objects: numbers, vectors, functions, matrices, quaternions etc. But we restrict our study to only those objects which follow a properly laid down axioms (a total 10 of them). It’s like a whole algebraic analysis (not in the mathematical sense of the word) of all possible mathematical objects (often called vectors) which follow those axioms. However, there are some books which treat linear algebra only as a tool of solving linear equations (of which I’m a very strong opponent). In a few books, you might find the whole effort is on solving and analysing ##A x =b ##, I feel that’s not a very good motivation.
Then there are books which treat the subject solely from matrix point of view, which is fine, but not good if you’re beginning for the first time.
Prof. Gilbert Strang and Mr. Tom Apostol teach the subject, as far as it seems to me, for the sake of the subject only. Prof. Strang focuses more on matrices, while Mr. Apostol always consider matrix as a representation of an operator.
How to study the subject?
Accept this fact that we don’t understand Linear Algebra in the beginning, we find no motivation for defining the axioms for a space, linear span and independence shall be seemed as a made-up thing, and a set is orthogonal and the same set could be not orthogonal (depends on how we define inner product) would seem a little unusual. But just remain with it for a month, and you might realize that you belong to a different class of people in society, we care about things that do not exist at all.
Linear Algebra and Quantum Mechanics:
If your final aim is QM, and if we are teleological people, completing Linear Algebra completely before entering into QM would be a bad idea. You might complete Linear Algebra, and then try to see QM as a mathematical subject, following some given axioms, by studying, for example, Mr. Dirac or Mr. Neumann; but this approach is not advised by learned men. Another possible pathway could be to try a few lectures of Leonard Susskind’s on YouTube, or/and Feynman’s Vol III, and see how much Linear Algebra you need to study before it and how much can be learned through it.
