Linear Algebra Proofs and Problems

In summary, there were some problems and proofs in a pdf that could be downloaded by anyone, but due to size limitations, a link to google docs was provided. However, the first link cannot be edited and errors must be corrected in the thread. The second pdf contains more advanced proofs. Some problems still need solutions and the rest should be checked. Additional comments and questions can be posted in the provided link. A review of the first document highlighted a few minor issues, including a typo and a mistake in reasoning for one problem.
  • #1
Dustinsfl
2,281
5
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
Theorems/Proofs Undergraduate level

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:

http://mathhelpboards.com/commentary-threads-53/commentary-linear-algebra-proofs-problems-4230.html
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
For the first document, here is what Ackbeet$\equiv $Ackbach suggested on MHF that needed to be adjusted
Ackbeet said:
Very nice review! I just had a few comments: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!
 

Similar threads

  • Calculus and Beyond Homework Help
Replies
24
Views
4K
  • Differential Geometry
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
3K
  • Calculus and Beyond Homework Help
Replies
3
Views
7K
Back
Top