# Math Help - Lin Alg Proofs and Counterexamples

1. There is a new updated file loaded with corrections. Numbers 3 and 4 under test 5 only have whether they are true or false and are missing appropriate proofs or counterexamples.

I would appreciate if some members could help with those questions. Also, if you spot any other questions that should be re-examined, please let me know.

2. The question under Test 5 number 3 is the only question left to go. It states similar matrices have the same eigenvectors.

This is false. Similar matrices have the same eigenvalues. Does anyone have a counter example or definition I may use to show this?

Example found.

3. Here is a counter example:
$
\begin{bmatrix} 7 && 2\\-4 && 1 \end{bmatrix} \: similar \: to \: \begin{bmatrix} 5 && 0 \\ 0 && 3 \end{bmatrix}$

if we look at the eigen value $\Lambda = 3$ then our eigen vectors will look like

$
\begin{bmatrix} 7 && 2\\-4 && 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3x \\ 3y \end{bmatrix}$

for which the solution, and the corresponding eigen vectors are
$
Ker(\begin{bmatrix} 4&& 2\\-4 && -2 \end{bmatrix})$

which are vectors of the form $\begin{bmatrix} \frac{-y}{2}\\y \end{bmatrix})$

where y can range anywhere over the field.

and for our similar matrix
$
\begin{bmatrix} 5 && 0\\0 && 3 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3x \\ 3y \end{bmatrix}$

f we look at the eigen value $\Lambda = 3$ then our eigen vectors will look like

$
\begin{bmatrix} 5 && 0\\0 && 3 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3x \\ 3y \end{bmatrix}$

for which the solution, and the corresponding eigen vectors are of the form
$\begin{bmatrix} 0\\y \end{bmatrix}$

where y can range anywhere over the field. clearly the soln space for the first one is is not identical to the soln space of the second one for the same eigen value. so their eigen vectors are not the same. for concreteness take, $\begin{bmatrix} -2\\4 \end{bmatrix}$ see that this vector is an eigen vector for one and not for the other.

4. I was able to find a matrix to show that the eigenvectors aren't the same. If you have a moment, can you look at the problems in the file to see if there are any errors?

Thanks.

5. checked it, looks good to go.

6. I added another proof, that tonio helped with back in April, to the file.

7. I have added Vector Space Axioms, Subspace Axioms, Inner Product, and Normed Linear Space to the file.

8. Originally Posted by dwsmith
I have added Vector Space Axioms, Subspace Axioms, Inner Product, and Normed Linear Space to the file.
I'll give it another 36 hours and if there's been no further changes I'll sticky the file.

9. Originally Posted by mr fantastic
I'll give it another 36 hours and if there's been no further changes I'll sticky the file.
I have stuck a copy of post #1 to this subforum. Any future comments, suggestions, erratum etc. should be made there. Please do not reply with thankyou posts. If you want to thank dwsmith, click on the Thankyou button at that thread.