@AmirAsghari : tell him to learn some programming, applied math, or something people will actually pay him to do when he grows up. There is a glut of math Ph.D.'s with no job skills to do anything but teach. The percentage of them that find rewarding employment is determined by market forces, no matter now brilliant they are.

BTW, I just asked a very similar question at math.stackexchange.com/questions/549570/…

I would recommend against introducing vector spaces at the start. In a perfect world, all students would be skilled at and interested in math and this would be the right way to do it. But the world is not perfect, and if you do this, your students will dislike you and your course from the start. I would recommend starting with some matrix stuff, like solving systems of linear equations, multiplying matrices, and the like before you hit them with the abstract stuff. Then you can use some matrix/vector stuff as examples to help the students understand the abstract stuff.

I thought the presentation of abstract material such as subspaces and inner product spaces was weak and relied excessively on matrix algebra. I know most students who aren't math majors hate this stuff and I don't know if there is any book that will make them like it.I admit I can't recommend another book. I used Strang's book once and, to put it positively, I'll say I much preferred Lay's (the only relative advantage is that Strang does cover the matrix exponential).

Presumably, since the proof requires projections, which are, I think, only defined for closed subspaces, Lay proves the Cauchy-Schwarz inequality only for subspaces of $\mathbb{R}^n$, though it is easy to prove for any inner product space. This is my opinion, but the definition of "vector space" includes unnecessary "closure" rules, which makes the intimidating list of rules even longer. The exponential of a matrix does not appear in the book at all. Consider that the book contains several semesters' worth of material, from what I have heard, this is a serious omission.

Lay has some serious flaws. He calls the dot product of two vectors in $\mathbb{R}^n$ "the inner product", as though this were the only inner product. I have gathered that this usage is common in the applied math world, but it is inappropriate in an introductory linear algebra book because you might want the students to learn the correct meaning of "inner product". The Cauchy-Schwarz Inequality is proven using projections, which is absurd, because all you need is some algebra and basic properties of inner product. The proof (using projections) is also more difficult than the usual proof.

@arsmath : I am not an expert in the rules for MathOverflow, but I think that if there is any doubt, you should try MathStackExchange first. This may be a "research level question", but a lot of very smart and knowledgeable people post at MSE and I am pretty sure you would get a good answer there. If you don't, then come back here. If you post at both sites simultaneously, you should state so in both places, with links to the other sites.