Let a1 an Rn and let A be the matrix whose jth column is aj

Let a_1, ...., a_n R^n and let A be the matrix whose j-th column is a_j. If A is an invertible matrix, prove that for every choice of b_1, ...., b_n R^n there exists a linear transformation T from R^n into R^n such that T(a_j) = b_j.

Solution

Since a₁,a₂, ...,a_n Rⁿ, therefore each a_i is a n-vector and hence A is a n x n matrix. Let us define T: Rⁿ Rⁿ : T(x) = A^T(x) where x is also a n-vector. Since A is invertible, therefore A^T is also invertible. Thus, if A^T(x) = b_n, then x = (A^T)^-1b_n = (A^-1)^Tb_n. Thus, T : Rⁿ Rⁿ is defined as T ( (A^-1)^Tb_n) = b_{n ,} is the required linear transformation.