Thread: Orthogonal projection on the line y=x?

Hi I need help with a linear algebra problem please.

Let T:R^2->R^2 be the orthogonal projection on the line y=x.

(a) Find a formula for T(x,y)
I don't know where to start on this one because I don't know how to define the transformation.If anyone could explain the transformation and process to find the formula it would be greatly apprerciated.

(b)Use the formula in (a) to show that T^2=T.
Here I just want to know whether T^2 is the composition of T with T?

(c) Find the eigenvalues an bases of the eigenspaces of T.
Here I just need to know how do I set up the matrix whose eigenvalues I must find?

Any input would be greatly appreciated.

Thanks in advance.

Originally Posted by chocaholic

Hi I need help with a linear algebra problem please.

Let T:R^2->R^2 be the orthogonal projection on the line y=x.

(a) Find a formula for T(x,y)
I don't know where to start on this one because I don't know how to define the transformation.If anyone could explain the transformation and process to find the formula it would be greatly apprerciated.


To orthogonally project a general vector into a line with basis you must map .

In your case, you can choose and get the general form from the above. Of course, the dot denotes the usual inner product in


(b)Use the formula in (a) to show that T^2=T.
Here I just want to know whether T^2 is the composition of T with T?


Yes, and when you'll get the form for it'll be obvious that

Tonio

(c) Find the eigenvalues an bases of the eigenspaces of T.
Here I just need to know how do I set up the matrix whose eigenvalues I must find?

Any input would be greatly appreciated.

Thanks in advance.

.

Hi again.

I understand the orthogonal projection part as that is the formula, but why can (a,b) be (1,1)? Since we're working in R^2 shouldn't we have 2 basis vectors like (1,0) and (0,1)? And once we have the new vector/s is/are that T? How does this relate to x=y? unless (1,1) is a basis of x=y if so then how do we confirm this?

Thanks for the help.

Originally Posted by chocaholic

Hi again.

I understand the orthogonal projection part as that is the formula, but why can (a,b) be (1,1)?


It has to be a basis for the subspace on which we're going to project, and that's why (1,0) and (0,1) don't work.

Tonio


Since we're working in R^2 shouldn't we have 2 basis vectors like (1,0) and (0,1)? And once we have the new vector/s is/are that T? How does this relate to x=y? unless (1,1) is a basis of x=y if so then how do we confirm this?

Thanks for the help.

.

Ok so I understand why it's (1,1) and finding the orthogonal projection results in the formula:
T(x,y)=T((x+y)/2,(x+y)/2)
Is this correct? How do you set up a matrix of this formula?

Originally Posted by chocaholic

Ok so I understand why it's (1,1) and finding the orthogonal projection results in the formula:
T(x,y)=T((x+y)/2,(x+y)/2)
Is this correct? How do you set up a matrix of this formula?

The formula is , and then

Tonio

O I get it because its 1/2x+1/2y and 1/2 is the coefficient.
Thanks a million. This really cleared things up for me!