Questions:
Linear Operators : Finite-dimensional Vector Space, Fields and Mappings
Let V be a finite-dimensional vector space. The base field F may be either R or C here. Let T, an element of the linear mapping of V to V, L(V), be an operator.
Suppose that all non-zero elements of V are eigenvectors for T. Show that T is a scalar multiple of the identity map, i.e., that there is a λ in the Reals such that T(v) = λv for all v in V
Clues that were given:
Assume we have 2 non-linear dependent vectors . . .
Means space is 1-dimensional . . .
If there are no 2 linearly independent vectors (i.e. dimV = 1) then . . .
If v and w are linearly independent, let Tv = λv and let Tw = µw. Want to show that λ=µ. We also have T(av +bw) = aλv + bµw for every a,b in the base field,F