Axler's Linear Algebra Done Right has always been my favorite linear algebra text (I say this 10+ years after reading it and finishing my math PhD, if that matters).

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