You might note that you can generate a -invariant subspace by taking a vector and taking the basis .
In R^2 with the standard dot product, define T(x,y)=(2x+y,0). Find a subspace U of R^2 such that U is T-invariant but U orthogonal is not T-invariant. Does T fix all the vectors in the subspace U you found?
I'm not really sure where to even start with this problem, any help would be appreciated!! thank you.