# Thread: Help with change of basis proof

1. ## 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.

2. 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 $L(V)$ 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. $STS^{-1}=T\Longleftrightarrow ST =TS\,,\,\,\forall \,S\in L(V)$ (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