If V is a finite-dimensional vector space, then the matrix A of a linear transformation is invertible if and only if T is one-to-one.

Question

If V is a finite-dimensional vector space, then the matrix A of a linear transformation  is invertible if and only if T is one-to-one.

Quizplus · Accepted Answer

This statement is known as the invertible matrix theorem for linear transformations, which states that a linear transformation (or matrix) is invertible if and only if it is both injective (one-to-one) and surjective (onto). Since a matrix is invertible if and only if its corresponding linear transformation is invertible, the matrix A of a linear transformation T is invertible if and only if T is one-to-one.

If V Is a Finite-Dimensional Vector Space, Then the Matrix