Richard Courant. Several years before I started studying mathematics in earnest, I spent a summer working through his calculus texts. Only recently, on re-reading them, have I come to realize how much my understanding of calculus, linear algebra, and, more generally, of the unity of all mathematics and, to use Hilbert's words, the importance of "finding that special case which contains all the germs of generality," have been directly inspired by Courant's writings.
From the preface to the first German edition of his Differential and Integral Calculus:
My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptuous purism; this is not the least of my purposes in writing this book.
Another example: while not a "linear algebra book" per se, I have yet to find a better introduction to "abstract linear algbera" than the first volume of Courant's Methods of Mathematical Physics ("Courant-Hilbert"; so named because much of the material was drawn from Hilbert's lectures and writings on the subject). His one-line explanation of "abstract finite-dimensional vector spaces" is classic: "for n > 3, geometrical visualization is no longer possible but geometrical terminology remains suitable."
Lest one be misled into thinking Courant saw "abstract" vector spaces as "$mathbb{R}^n$ in a cheap tuxedo," he introduces function spaces in the second chapter ("series expansions of arbitrary functions"), and most of the book is about quadratic eigenvalue problems, or, as Courant saw it, "the problem of transforming a quadratic form in infinitely many variables to principal axes."
As a final example: Courant's expository What is Mathematics? is perhaps best described as an unparalleled collection of articles carefully crafted to serve as an object at which one can point and say "this is." Moreover, while written as a "popularization," its introduction to constrained extrema problems is, without question, a far, far better introduction than any textbook I've ever seen.
I should also mention Felix Klein, not only because Klein's views on "calculus reform" so clearly influenced both the style and substance of Courant's texts, but since a number of Klein's lectures have had an equally significant influence on my own perspective. For those unfamiliar with the breadth of Klein's interests, I'm tempted to say "his Erlangen lecture, least of all" (not that there's anything wrong with it).
Lest my comments be mistaken for a sort of wistful "remembrance of things past," I'd easily place Terence Tao's writings on par with Courant's, for many of the same reasons: clear and concise without being terse, straightforward yet not oversimplified, and, most importantly, animated by a sort of — je ne sais quoi — whatever it is, it seems to involve, in roughly equal proportions: mastery of one's own craft, a genuine desire to pass it on, and the considerable expository skills required to actually do so.
Finally, I can't help but mention Richard Feynman in this context, and to plug his Nobel lecture in particular. While not a mathematician per se, Feynman surely ranks among the twentieth century's best examples of a "mathematical physicist" in the finest sense of the term, not merely satisfied by a purely mathematical "interpretation" of physical phenomena, but surprised, excited, and, dare I say, delighted by the prospect! Moreover, he was equally excited about mathematics in general, see, e.g., the "algebra" chapter in the Feynman Lectures on Physics.
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