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.