If is a linear transformation, with V a vector space having basis , and if for all i, where is a scalar, then the matrix of T is diagonal, where G is the basis used for both the domain and codomain.

Question

If  is a linear transformation, with V a vector space having basis  , and if  for all i, where  is a scalar, then the matrix of T is diagonal, where G is the basis used for both the domain and codomain.​

Quizplus · Accepted Answer

By the given conditions, the linear transformation T maps each basis vector of V to a scalar multiple of itself. This implies that the matrix of T with respect to any basis of V (including the given basis) is diagonal.

If Is a Linear Transformation, with V a Vector