Dimension of Eigenspace of A and A^T

[potassium_mn04]

Homework Statement

Prove that the dimension of the eigenspace of A and dimension of eigenspace of A^T are equal

Relevant Equations

dim(E_A)=dim(E_(At))

I know that the rank of A and A^T are equal, and that the statement follows from there, but I have no idea how to prove it.

 

[S.G. Janssens]

The eigenspace of $A$ corresponding to an eigenvalue $\lambda$ is the nullspace of $\lambda I - A$. So, the dimension of that eigenspace is the nullity of $\lambda I - A$. Are you familiar with the rank-nullity theorem? (If not, then look it up: Your book may call it differently.) You can apply that theorem here.