Trivial solution equivalent to a solution existing

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.