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?