Thread: Help with change of basis proof

Let V be a finite-dimensional vector space over F, and suppose that T ∈ L(V ) is a linear operator having the following property: Given any two bases b and c for V , the matrix M(T, b) for T with respect to b is the same as the matric M(T, c) for T with respect to c. Prove that there exists a scalar α ∈ F such that T = αidV , where idV denotes the identity map on V .

I don't really know how to approach this, and some guidance would be helpful! Thanks.

Originally Posted by blackwrx

Let V be a finite-dimensional vector space over F, and suppose that T ∈ L(V ) is a linear operator having the following property: Given any two bases b and c for V , the matrix M(T, b) for T with respect to b is the same as the matric M(T, c) for T with respect to c. Prove that there exists a scalar α ∈ F such that T = αidV , where idV denotes the identity map on V .

I don't really know how to approach this, and some guidance would be helpful! Thanks.

Let's see if you can follow me: any element in can be seen as a change of basis transformation, since a linear map is invertible iff it maps a basis into a basis, so what the problem is giving is that T is a linear map s. t. (why?).

What's left now is to prove that the only lin. maps that commute with ALL the elements in L(V) are the scalar multiples of the identity map, and this is a nice, fairly non-hard exercise. One way to approach it is to work with matrices...

Tonio