February 09, 2020
From Axler's Linear Algebra Done Right, chapter 3.D, exercise 20:
Suppose is a positive integer and for . Prove that the following are equivalent. (note that in both parts below, the number of equations equals the number of variables):
(a) The trivial solution is the only solution to the homogenous system of equations
(b) For every , there exists a solution to the system of equations
The left hand side of both system of equations can be rewritten as the matrix equation , where and . The matrix induces the linear map where .
Condition (a) is equivalent to , which is equivalent to is injective. Condition (b) is equivalent to , which is equivalent to is surjective.
Hence, [Axler 3.69] implies that injectivity and surjectivity are equivalent for linear operators, and as , we have injective iff surjective. Therefore, (a) and (b) are equivalent.