Finding a teacher isn't at all mandatory, you can do math yourself by completing exercises, checking your work, and looking at the solution afterwards. There are plenty of places to ask questions if you get stuck.
Axler's approach for determinants.
Say what?
This is probably the correct generalization of the volume definition.
You don't start kids with complex numbers until they can handle reals. You don't even start negative numbers until they can handle positive numbers.
The only beneficial traditional treatment I can think of is saying "determinant is volume", and that's what Axler does.
From there, one can look into alternating forms, convince oneself that an alternating n-linear form does the same thing, and obtain a formula for it (e.g. summing up signed products of numbers on the diagonals over all permutations of columns).
If by "traditional" you mean: "Here's an insanely complicated formula that does something magical. Learn to compute it. On page 5, we'll prove that it tells something about independence. Oh, and we'll mention volume on page 10 briefly" -- then I not only think this is not useful, I think it's outright harmful.
Axler defines determinant as (up to a sign) the constant term of the characteristic polynomial, and he needs two different definitions for characteristic polynomial, one over R and one over C. Now what if the ground field is something else? Do we need yet another definition of characteristic polynomial in order to define the determinant? What if you are doing linear algebra over a commutative ring?
LADR actually presents a very narrow view about linear algebra : it treats linear algebra merely as finite-dimensional functional analysis. The readers can be hit hard when they need to do other (computational or theoretical) stuffs. Similar concerns had been voiced on the internet before. In particular, I think Darij Grinberg's comments (below the answer https://mathoverflow.net/a/16996) on LADR are rather spot on.
It's fine if you find LADR helpful. The book does have its merits (I like its clear and fluent writing and its neat proofs), but it has also its own shares of problems and there are other nice choices of books in the wild.
(I'm more familiar with this phenomenon in philosophy, where the greater the philosopher, the more they have entirely their own way of looking at things, untranslatable into another tongue, which you just have to come to understand on its own terms. A summary of their views leaves out the personal aspect, the style, the way of thinking, and will seem dead.)
I firmly disagree.
What a teacher can give you is perspective that you, not knowing the subject, cannot have -- and augmenting any particular viewpoint expressed in a book.
If people could learn everything from texts, we wouldn't have universities (for students) and conferences (for working mathematicians).
In theory, one can write a Great Text that explains an Idea. In practice, it's damn hard to do that, and it's far easier to impart understanding in a conversation, filling in any blank spots the audience might have on the spot, and guiding the way in the jungle.
That's why all texts are kind of bad. Either they are too narrow to give a wide perspective, or too huge to be absorbed!
As one of my advisors said: mathematics, like food, is best shared. Don't go into it alone; and whether you have or don't have a mentor, try to find someone else to join you on your journey (a friend who wants to learn the same subject).
Also I recall reading Axler, and I think it covered some more advanced content towards the end. But this book looks equally solid.
You could solve the problem with something like a wiki but I've seen that usually results in quality problems. It isn't like a wiki article where all the results can basically see what has been written. With a wikibook though...you need all writers to basically know where stuff has already been explained...otherwise you just get a bunch of people explaining x here and then other people explaining the same thing in a later chapter.
Linear Algebra Abridged, a free compactified version of Linear Algebra Done Right, 2016. http://linear.axler.net/LinearAbridged.html
Linear Algebra Done Right videos, free videos to accompany the book, 2017. http://linear.axler.net/LADRvideos.html
For those who want a free alternative, behold: Treil's Linear Algebra Done Wrong[2]
The books is downloadable as a free PDF. The name is an answer to Axler's book (dry mathematician's humor), and offers an opposite approach (getting to determinants first).
While I agree with Axler and diagree with Treil, LADW offers way more examples and applications, and together LADR and LADW offer a complete, excellent course material.
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As for the linked text: not a bad text, but I wouldn't pick it over LADR + LADW.
Here's why:
1. size: it's larger than LADR+LADW taken together. It's hard to see the forest behind the trees.
2. exposition: it follows the structure of many other texts that I don't like because they terribly confuse the students (that I'd have to re-teach afterwards): starting with solving systems of linear equations, then jumping into vector spaces, for example.
3. I don't like how key concepts (matrix product, determinant are introduced). If you already know the material, it will be hard to see what's wrong with the approach of throwing a definition at the reader, and then talking about why that definition was made. But the opposite should be the case.
After teaching Linear Algebra, here's my litmus test for a good book. At a glance, it should make the following clear first and foremost:
1. A matrix of a linear map F is simply writing down the image of the standard basis F(e_1), F(e_2), ... F(e_n). These vectors are the columns of the matrix. If you know them, you can compute F(v) for any v by linearity. That's called "multiplying a vector by matrix"; we write Mv = F(v).
2. The product of matrices is simply the matrix of composition of linear maps that they represent. The student can figure out what that matrix should be (or should be able to do so); here's how. If M is the matrix of F, and N is the matrix of G (where F and G are linear maps), then the first column of MN is F(G(e_1)) = M x (first column of N). Same for other columns. Ta-dah.
3. The determinant of v_1, .. v_n is simply the volume of the lopsided box formed by these vectors (mathematicians call the box "parallelepiped"). In particular, in a plane, the area of the triangle formed by vectors A and B is half the determinant. This are can have a minus sign; switching any pair of vectors flips the sign.
4. Eigenvectors and eigenvalues are fancy words that allow us to describe linear maps like this: "Stretch this picture along these directions by this much". Directions are eigenvectors, by how much - eigenvalues.
Bonus:
5. Rotation and scaling are linear maps. That's all any linear map does: rotates and stretches. Writing a map down in this way is called singular value decomposition.
6. Shears are linear maps that don't change the volume. Any box can be made rectangular by applying a bunch of shears to it. That's called Gaussian elimination or row reduction when you look at what happens to matrices (and apply scaling as the last step). This is also an explanation of why the determinant gives volume (if you define it as an alternating n-linear form).
That's the beginning of a solid understanding of the subject.
From my experience, LADR+LADW leave the student with an understanding of 1-4, and other texts, due to being organized badly, don't (even when they contain all the information in some order).
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Books I recommend:
[1]LADR: http://linear.axler.net/
[2]LADW: https://www.math.brown.edu/~treil/papers/LADW/LADW.html
Just to nitpick, but this sentence might be read ambiguously by some people, who would understand it as the definition of shear. You may want to say "Shears are an example of ...".