orthonomal basis

Kat-M · #1 $Old$ January 12th, 2009, 03:48 PM

Let T be a linear transformation from R^2 to R^2. and T is represented by the matrix B=
[(1/5)^(1/2) -2(1/5)^(1/2)]
[2(1/5)^(1/2) (1/5)^(1/2)]
with respect to the stantard basis of R^2.

a) Is T isometry?
b) does R^2 have an orthonormal basis for eigenvectors of T?

i can do a) but need help on b). i dont really understand what it is asking. is it asking to find an orthonormal basis for eigen space of T?

PaulRS · #2 $Old$ January 13th, 2009, 05:45 AM

Quote:

Originally Posted by Kat-M $View Post$

Let T be a linear transformation from R^2 to R^2. and T is represented by the matrix B=
[(1/5)^(1/2) -2(1/5)^(1/2)]
[2(1/5)^(1/2) (1/5)^(1/2)]
with respect to the stantard basis of R^2.

a) Is T isometry?
b) does R^2 have an orthonormal basis for eigenvectors of T?

i can do a) but need help on b). i dont really understand what it is asking. is it asking to find an orthonormal basis for eigen space of T?

It asks whether there's an orthonormal base for $\mathbb{R}^2$ made out of eigenvectors of $T$

What they want you to see is that, if you were working with complex numbers instead of real numbers, b) would hold -by the spectral theorem for Unitary Transformations-, but here it may not ( Try !

).