[College - Linear Algebra] Homework help

[Jtehanonymous]

Hey everyone. I was wondering if I could have some help with two questions I'm currently stuck on. I was a little unsure whether I should post in this forum since it's basically dead...but I was hoping someone would find their way in here and help out. xD

1) Let A be a 7x5 matrix of rank 4. Let P and Q be the projection matrices which project vectors in R^7 onto R(A) and N(A^T). Find the inverse of P-Q.


2) What is the closest function, a cos(x) + b sin(x), to the function f(x) = sin(2x) on the interval from 0 to pi?


Any help or ideas at all would be greatly appreciated. Thanks!

[emerald000]

I can't help you for the first one, here's how to do the second one:

First you'll need to define what is the "closest" function. I guess it will be the integral of the differences of the absolute values of the functions from 0 to π. It can still be applied to other definitions though. Once you have your function defining that difference, all you will need to do is an optimization problem to find the global minimum.

I hope it makes sense.

EDIT: Wow. Pi is awful in this font.

[iironiic]

Originally Posted by Jtehanonymous

1) Let A be a 7x5 matrix of rank 4. Let P and Q be the projection matrices which project vectors in R^7 onto R(A) and N(A^T). Find the inverse of P-Q.


2) What is the closest function, a cos(x) + b sin(x), to the function f(x) = sin(2x) on the interval from 0 to pi?

For the first question, I don't understand what is N. Is it a matrix?

EDIT: Actually, can you explain the entire first question? I don't really understand it xD

I'll look into the second one.

[emerald000]

Good to know I am not the only one not understanding the first question.

[Jtehanonymous]

I'll try to explain the symbols in number one more.

R(A) represents the Range of Matrix A and N(A^T) is the Null space of A transpose. These are orthogonal compliments and when you multiply vectors from these bases together you get 0.

[iironiic]

Originally Posted by Jtehanonymous

I'll try to explain the symbols in number one more.

R(A) represents the Range of Matrix A and N(A^T) is the Null space of A transpose. These are orthogonal compliments and when you multiply vectors from these bases together you get 0.

Can you take the null space / range of a matrix? I thought you can only do that with the transformation the matrix is associated with. Is that what you are implying?

[Jtehanonymous]

Originally Posted by iironiic

Can you take the null space / range of a matrix? I thought you can only do that with the transformation the matrix is associated with. Is that what you are implying?

R(A) is the columns that contain lead variables when you reduce A to simplest row echleon form.

N(A^T) is solving the equation (A^T)x = 0. That is, transposing A and then setting this system equal to the trivial solution.


These solution sets form the bases for R(A) and N(A^T).

[SocoNhydro420]

Originally Posted by iironiic

Can you take the null space / range of a matrix? I thought you can only do that with the transformation the matrix is associated with. Is that what you are implying?

Well, linear transformations have matrix representations. So yes the range and nullspace can be taken for matricies.

[stargroup100]

I do realize how late I am to this discussion, but there is a very elegant solution to the second problem that I would like to point out.

Hint: angle sum and difference trigonometric identities