In sincere disagreement with the negative reviews here, I absolutely loved this book. I took a class on linear algebra my freshman year of college that used Otto Bretscher's book, and I read it pretty much from cover to cover. Now, I wasn't a very studious student at the time, so this says a lot about just how well Bretscher presented the subject -- it was so interesting that I just wanted to keep reading. Unlike more advanced textbooks in math, Bretscher's book is rigorous while remaining friendly. I was always assured that I could understand anything in the text after a few tries and a bit of thinking, because enough detail is included to make sure the engaged reader does not get lost. The result is a highly streamlined presentation that makes you really appreciate the beauty of linear algebra, and takes you through some cool theorems, like the Spectral Theorem.
Other reviewers have remarked that the writing is "terse" or complained that examples are "done in symbols". In response to the former complaint, I don't think the explanations are terse at all. Sure, the text might make you think more and work out more things on your own than any math book you've encountered before reading this one, but it's certainly very doable for those who aren't lazy. In fact, in comparison to an often-used math book (though not linear algebra) like Rudin's Principles of Mathematical Analysis, Bretscher's book helps the reader out way more. So much more. The text is just really clear and unintimidating. I can't emphasize this enough.
In response to the complaint about examples, I found the examples to be very accessible, straightforward and illustrative of the concept at hand. They're well-integrated and, yes, they're done in math symbols, but, well, what else could you expect from a math book? Sure, you could try to explain concepts verbally instead, but usually it's just better to be walked through a "symbolic" example and see how the math is really done. After all, as a learner of the subject, you'll have to do it yourself as well.
And the problems are great. They can definitely be solved using knowledge gained from the relevant section in the text, though not all are completely straightforward, as previous unsatisfied reviewers seem to be expecting. I say this with such confidence because I did many of the problems myself without knowing anything about linear algebra that Bretscher's book did not teach me.
Since taking linear algebra, I've had to recall concepts I'd learned and use them in other science and math courses, and to my surprise I still have a good grasp of the important ideas, no doubt due in large part to the fine teaching of the book being reviewed.