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I'm looking for suggestions for linear algebra books, suitable for first year students, that emphasize linear transformations. Rigorous, with proofs and every detail, ideally starting out with concrete vector spaces, immediarely introducing linear transformations with lots of examples, a bit about linear subspaces, and then developing the subject under the motto "linear algebra = linear transformation theory", leading to abstract ideas and the usual material all presented as means to study linear transformations. Books along similar lines would be of interest.

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You might want to take a look at Linear Algebra Done Right. The Table of Contents and other information about the book are available at www.linear.axler.net. Linear Algebra Done Right very much emphasizes the linear transformation approach, so it might be suitable for you.

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    Thank you! I'm ordering the book. – 2017-02-04
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My first semester in college I took a course for which the textbook was written by the instructor himself (then-chair of the dept.), available here.

Though a bit abstract in its exposition, there are lots of examples. Most of the theorems leave proofs as exercises for the reader, as per the Moore method; but various hints, templates and explanations as to how previous theorems may be used to construct a given proof are provided. So, though merely reading it will not improve one's understanding, it is a pleasure to work through, at least in my experience.

I hope that helps.

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    Thank you for your suggestion, but unfortunately this is precisely what I was not looking for. Linear transformations make first appearance only in chapter 7. I'm looking for a text that emphasises linear transformations as the reason for linear algebra, particularly to counteract the prevalent approach of quickly going over to matrices, just because they are easy to define, giving an impression as if somehow mathematicians are obsessed with rectangles of numbers. Again, thanks though for your input. – 2017-02-03