Let be a field, and let and be – and -dimensional vector spaces over , respectively. Prove that, for any bases and , is a nonsingular linear transformation if and only if is nonsingular matrix.
Choose some basis and .
Suppose is a nonsingular linear transformation. Then . Now suppose . If , then , and so . Thus is a nonsingular matrix.
Conversely, suppose is a nonsingular matrix. Suppose . Then , and so we have . Since is nonsingular, . Thus , and so is nonsingular as a linear transformation.