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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?

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  • $\begingroup$ I would say that the subject matter you refer to as a whole is a marvel of efficiency and beauty: the basic material you describe can be done in about five pages. In my opinion the one result which makes it all beautiful and easy is the Steinitz Exchange Lemma. To me the basic dichotomy of exposition of this material is whether SEL is soft-pedaled/avoided (e.g. by row reduction arguments) or explicitly embraced, and when the latter is done you get something beautifully simple. It is also possible to treat this material in a more abstract context, leading to the notion of matroids. $\endgroup$ – Pete L. Clark Jan 28 '14 at 5:43
  • $\begingroup$ On the other hand, I am not sure that I have seen any two presentations of this material that I would regard as "essentially different". $\endgroup$ – Pete L. Clark Jan 28 '14 at 5:45
  • $\begingroup$ I think Lax's linear algebra book is mathematically beautiful. $\endgroup$ – littleO Jan 28 '14 at 5:48
  • $\begingroup$ @PeteL.Clark Could you recommend a resource that presents the material using the Steinitz Exchange Lemma result? $\endgroup$ – user89 Jan 28 '14 at 5:52
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    $\begingroup$ @twirlobite: I think that this is done in most of the more "theoretical" approaches to linear algebra. I don't really have a go-to linear algebra text, I'm afraid. But e.g. these notes give the approach that I have in mind:web.math.princeton.edu/~mdamron/teaching/S13/notes/…. $\endgroup$ – Pete L. Clark Jan 28 '14 at 6:41
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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.

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  • $\begingroup$ Could you please share a little about why you find the presentation elegant? I have edited my question as well to ask for this in future answers. $\endgroup$ – user89 Jan 28 '14 at 5:24
  • $\begingroup$ @twirlobite: I edited to put as much info that I remember; my general recollection is that I jumped in joy several times while reading the book, or trying to do the exercises. They helped me a lot to get a clear picture of linear algebra. $\endgroup$ – voldemort Jan 28 '14 at 5:31
  • $\begingroup$ I came across the following comment in an Amazon review "On the other hand, there is little about linear mappings between vector spaces of different dimensions, which are crucial for differential geometry." What do you think about that analysis? $\endgroup$ – user89 Jan 28 '14 at 5:33
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    $\begingroup$ @twirlobite: I wouldn't fully agree. He doesn't treat them at great details, but then the results can be generalized pretty easily, by what he chooses to say, and the previous knowledge. Anyway, I didn't have any trouble even taking a graduate level differential geometry course; and my knowledge in Linear Algebra came from this book. $\endgroup$ – voldemort Jan 28 '14 at 5:39
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    $\begingroup$ This last bit is interesting, it suggests that in math, one does not need a direct acquaintance with all of the preliminary or directly support material as much as a familiarity with the material in general and the mathematical maturity to both backfill the substantive or logical gaps in subsequent exposure to different and perhaps more advanced mathematics. $\endgroup$ – Erik G. Jan 28 '14 at 16:52

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