I have a quick question regarding the method of finding eigenvalues and eigenvectors with linear transformations.
Suppose I have a linear transformation T that has a matrix representation A with respect to some basis.
What would $I$ be in $$det(A - \lambda I)$$
Would it be simply the standard identity matrix or the basis vectors for the matrix representation of T?
It is just the identity matrix. That is, the matrix with $1$'s along the diagonal and $0$'s elsewhere.
Recall that this comes from starting with an expression of the form $Av = \lambda v$ and then manipulating this to $Av - \lambda v = 0 \implies (A - \lambda I)v = 0$. Where $I$ is the identity matrix. We see that the basis chosen to produce $A$ has no effect whatsoever on the introduction of $I$ into this expression.