I had not yet read it. Having just fixed it, it is missing the role of a basis. And without it, you'll have trouble figuring out coordinate systems, changing your basis, and so on. Which gives you no way to keep straight such confusing things as the fact that if your coordinate system rotates one way, your representation of things rotates the other. (Try it! Stand up, turn clockwise and see the world spin counter-clockwise!)

Let me offer an abstract example. Consider the polynomials of degree at most 2. There is an obvious basis, namely 1, x, and x^2. A polynomial like x^5 - 3x + 2 can now easily be written in coordinates as (2, -3, 1).

However we have many other coordinate systems that might be convenient. For example suppose that we're sampling data, and can measure p(0), p(1) and p(2). How do we find what polynomial that is? Here is an easy way. We can easily find the new coordinates for our basis vectors: 1 -> (1, 1, 1), x -> (0, 1, 2), x^2 -> (0, 1, 4). That means that we can write down the matrix representing the identity transform (nothing happened), going from the basis we have, to the new coordinate system:

    ( 1  1  1)
    ( 1  2  3)
    ( 1  2  4)

That's the change of basis matrix one way. Invert it.

    ( 2 -1  0)
    (-3  2 -1)
    ( 1 -2  1)

And now we can go the other way. The polynomial that we want from our sample data will be (2-3x+x^2)p(0) + (-1+2x-2x^2)p(1) + (-x+x^2)p(2).

There are a lot of problems where linear algebra comes up that you can think through more clearly if you think about things this way (complete with the role of the basis!) than if it isn't fully digested.

As for a better book, well, I already recommended Down With Determinants! :-)