Uni Maths Questions

[kamil9876]

So just evaluate Av where v is the column vector you have provided us. Each entry is in fact a sum of entries in a row (think about the definition of matrix multiplication.

[Alwin]

So just evaluate Av where v is the column vector you have provided us. Each entry is in fact a sum of entries in a row (think about the definition of matrix multiplication.

no no I was supposed to prove it first THEN HENCE find v, not the other way around. but dont worry, it's kool. I mucked around with determinants and the characteristic polynomial and got it eventually. it's not a very slick method tho.. Kamil9876 if u have a suggestion as to the proof (without using v first!) I'd love to hear it! or anyone else for that matter haha. thnx.

[Mao]

no no I was supposed to prove it first THEN HENCE find v, not the other way around. but dont worry, it's kool. I mucked around with determinants and the characteristic polynomial and got it eventually. it's not a very slick method tho.. Kamil9876 if u have a suggestion as to the proof (without using v first!) I'd love to hear it! or anyone else for that matter haha. thnx.

How is demonstrating that k is an eigenvalue not enough proof? Not every mathematical proof has to be a direct proof. This is an example of a proof by construction, which is equally valid. http://en.wikipedia.org/wiki/Proof_by_construction

[Alwin]

How is demonstrating that k is an eigenvalue not enough proof? Not every mathematical proof has to be a direct proof. This is an example of a proof by construction, which is equally valid. http://en.wikipedia.org/wiki/Proof_by_construction

That's the thing. I know proof by construction, but how do you propose to prove it here without using the matrix v ?

It's like trying to solve a quadratic like the following:



I construct the answer to be 2,3 Then,



Oh, so my construction is correct!!
-.-

Seriously, you almost always find the eigenvalues first then the eigenvector. So explain please how to use proof by construction to prove:

that if all the rows of a matrix add up to the same number k, then k is an eigenvalue of this matrix.

Sorry, it's not that I'm being uncooperative, it's just that I have this massive ugly looking proof for it and it seems wayy to long...
EDIT: Don't worry guys, I got it.

[Mao]

So explain please how to use proof by construction

Let A be a square matrix where each row adds up to k.

It follows , where is a column vector with each element being 1.

Therefore, k is an eigenvalue to A.

QED.

[Yendall]

Let be the predicate . If the domain of interpretation is , is the following true or false? There exist and such that is true.
When the universe of discourse has only two variables, how do I determine Z?

I might be wrong, but wouldn't it be via assumption? Like the pattern 1,2... would step to 3?

[kamil9876]

When the universe of discourse has only two variables, how do I determine Z?

But of course the do not have to be distinct! So you want to find such that . Take and you see that indeed is true. Hence the statement is true: there do exist such .

[Yendall]

But of course the do not have to be distinct! So you want to find such that . Take and you see that indeed is true. Hence the statement is true: there do exist such .

Ohhhhh! Thank you, makes so much more sense.

[Deleted User]

Let S: P2 -> P3 be defined as follows. For each p(x) = a2*x^2 + a1*x + a0, define S(p) = 1/3*a2*x^3 + 1/2*a1*x^2 + a0*x. Find the matrix A that represents S with respect to the bases B={1,x,x^2} and B'={1,x,x^2,x^3}.

How do I do this question thanks!!!!

[MJRomeo81]

Hey guys, is anyone in the mood for tackling some recurrence relations?If someone could provide a detailed solution that would be great.

http://i.imgur.com/tdJPbn8.png

[b^3]

Not going to stick to it exactly, but give you enough so that you should be able to do the question.

4. This is a homogeneous recurrence relation since we don't have any terms that don't involve S(something).

Charateristic Equation


 
This means our general solution will be of the for


 
Then the complete solution will arise when we fit the two initial conditions to the situation.

 
Can't quite get 5. to work, may give it another shot tomorrow, or someone else might be able to help.

[BubbleWrapMan]

Solution to Q5 attached.

[Deleted User]

How do I find the eigenvector of a 2x2 matrix?
So for
3 -2
2 -2

I've found that the eigenvalues are -2 and 1.

Then, for eigenvalue -2, I solve
|5 -2||x| = |0|
|2  0||y|    |0|

So 5x-2y=0 and 2x=0. How do I solve for the eigenvector from here? What do I let equal to the parameter?

Thanks

[Alwin]

How do I find the eigenvector of a 2x2 matrix?
So for
3 -2
2 -2

I've found that the eigenvalues are -2 and 1.

You may want to check that working. I get 2 and -1

So 5x-2y=0 and 2x=0. How do I solve for the eigenvector from here? What do I let equal to the parameter?
                Note these equations would also be incorrect
Thanks

And for your question, yes you do let a parameter, say t, equal y. Then let t equal a convenient number. t=1 is always pretty good!

[Deleted User]

Ok thanks. Does it matter whether I let x or y be the parameter?