Hello,
I have a question to do which I'm a bit stuck on. Any and all help would be much appreciated!
Assume V is an inner product space. If dim(V) = 3, for which of the following matrices does there exist a self-adjoint linear transformation on V for which the matrix represents with respect to some basis?
For the first one I'm thinking there's not, because of the row of 0's at the bottom. However, I'm still not 100% sure.
(I understand what a self-adjoint linear transformation is, and I also know that the eigenvectors of V will form an orthonormal basis of V.)
Thank you, dwsmith. From what I can tell, then, cannot represent a linear transformation, because it only has two eigenvectors, and so cannot form an orthonormal basis of V (2 vectors cannot span V). Does it remain, then, to check whether the eigenvectors of and form an orthonormal set?
Be careful, here, every matrix can represent a linear transformation. But in your post you said self-adjoint linear transformation. Every self-adjoint linear transformation has a "complete set" of eigenvectors- a set of independent eigenvectors equal to the size of the matrix. That means that the eigenvectors could be used as a basis for the space.
You also said " Does it remain, then, to check whether the eigenvectors of and form an orthonormal set? "
An "orthonormal" set consists of vectors that all have length 1 and are orthogonal. In fact, here, both sets contain a vector that does not have length 1 so these are NOT "orthonormal". All you need are that the vectors are independent.