# Linear Algebra Proofs and Problems

#### dwsmith

##### Well-known member
We used to have a bunch of problems and proofs that were in a pdf could be downloaded by anyone. Since we aren't able to upload pdf files of a certain size, I provided a link to google docs. If there is an error, typo, or something is just drastic wrong let me know.

Undgraduate Final Review Practice problems with solutions

However, with this first link, I can't edit this document. It was created with Maple which I no longer have. So errors have to just be corrected in the thread and then consolidated for readability.

This pdf has more advanced proofs in it.

Linear Alg Workbook

I have completed the second set. The only ones that need solutions are $A5$ part 2, $B7$ part2, $C4$ part 2 and 3, $C5$, $D4$, $F7$ needs to be checked, $H3$, $H10$, $I4$ part 3, $I5$, $I10$ part 2, $J7$, and $J10$.
The rest of the problems I believe to be right but they should still be checked out.

Comments and questions should be posted here:

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#### dwsmith

##### Well-known member
For the first document, here is what Ackbeet$\equiv$Ackbach suggested on MHF that needed to be adjusted

1. From Test 5, Problem 4, on page 4. I would say more than eigenvectors must be nonzero, by definition. It's not that the zero eigenvector case is trivial: it's that it's not allowed.

2. Page 6, Problem 8: typo in problem statement. Change "I of -I" to "I or -I".

3. Page 8, Problem 21: the answer is correct, but the reasoning is incorrect. It is not true that $\mathbf{x}$ and $\mathbf{y}$ are linearly independent if and only if $|\mathbf{x}^{T}\mathbf{y}|=0.$ That is the condition for orthogonality, which is a stronger condition than linear independence. Counterexample: $\mathbf{x}=(\sqrt{2}/2)(1,1),$ and $\mathbf{y}=(1,0).$ Both are unit vectors, as stipulated. We have that $|\mathbf{x}^{T}\mathbf{y}|=\sqrt{2}/2\not=0,$ and yet
$a\mathbf{x}+b\mathbf{y}=\mathbf{0}$ requires$a=b=0,$ which implies linear independence.

Instead, the argument should just produce a simple counterexample, such as $\mathbf{x}=\mathbf{y}=(1,0)$.

Good work, though!

• Ackbach