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Is there a specific question that you are looking for the solutions? I have partial solutions since I didn't do all the questions.
One of our OpenStudy members, @frankhecker , is working through the problems and blogging their solutions. I believe he is almost through with chapter 2.
Here's a link to his blog. http://math.hecker.org/\
That said, I've found working through the problems and asking questions on this forum is enough for me. Often, there are two ways to solve a problem, and sometimes I check my work by solving the problem two different ways. This gives a little extra practice and is sometimes helpful in understanding the material from different viewpoints.
btw, the hecker blog is very well done (from what I've seen). He shows his work and not just the solutions, sometimes offering a clear explanation than the brief ones in the solutions at the end of the book.
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@datanewb, thanks for the mention of my math homework blog. For others who are interested, there's a couple of things I should mention: First, I'm working through the exercises to the 3rd edition of Strang's "Linear Algebra and Its Applications", not the 4th edition, so the exercise numbers may not match up for those using the most recent edition. (I didn't feel like paying $100-200 for the 4th edition, and got the 3rd edition used for a really low price.)
Second, the blog is very much a spare time activity, and I don't have time to post much to it; converting one of my handwritten solutions to an actual blog post takes a while. Right now I'm trying to post one solution a week, and am having trouble keeping up with that schedule. So those who are expecting to see the solutions for the rest of the chapter 2 exercises may have to wait a while.