I am familiar with the standard proofs presented in textbooks for stuff like linear independence/dependence, the dimensions of common vector spaces, any basis for a vector space V must be linearly independent and have at least n = dim V vectors, etc.
However, I am curious to know this: are there books that present these proofs in (the most?) an elegant way? By elegant here, I am alluding to some intangible sense of: "beautifully simple", "a proof that presents a new way of looking at things", "using non-standard methods to form a particularly straightforward argument", etc.
Perhaps these proofs have some quality akin to 'breathtaking' to students familiar only with the standard presentation, or perhaps they convincingly demonstrate the power of particular branch of mathematics?
In your answer, could you share a little as to why you consider the presentations you are advocating elegant?
I loved Halmos's Finite dimensional vector spaces for its elegance. What I loved most about the book was that ideas were all well strung together that the whole book is like a garland of pearls, and not just a dazzling collection of them.
I loved the part where he proved that an n-dimensional vector space is isomorphic to $F^n$ where $F$ is the base field, and then proceeds to explain why we still need to study finite dimensional vector spaces abstractly. He introduces all main concepts pretty easily, and early, like the concepts of dual space (and some of the aspects simplifies the proofs). I don't remember all the details, but I do remember that I loved following his proofs; all of them were elegant.
His Linear algebra problem book contains great problems too.