The Archive

To see problems and solutions in the fall series, which runs from October 13 through December 15 visit The Fall 2025 Featured Problem Series

Problem

Our problem this week is from Penn State Math 436, an upper level linear algebra course. This course is typically the first time that students experience the full power of abstraction in mathematics, however, in a context that is not too far removed from the concrete. Learning linear algebra at this level is like learning to swim in deep water by swimming in a pool where your feet can touch the bottom.

The choice for this weeks problem is one which illustrates the power of abstraction.

Suppose that and are finite vector spaces over a field , which may either be or . Prove that there exists an injective linear map from to if and only if .

Solution

Four theorems from linear algebra are used in the solution of this problem. These are, in the order of appearance in the proof:

and

First, it is proven that the existence of a injective map implies that Suppose that there exists an injective . The first result above gives , which implies that . Since is finite-dimensional, The Fundamental Theorem for Linear maps is applicable. Thus By definition, is a subspace of , which is finite-dimensional by assumption. Therefore, by Theorem , The equality and the inequality are combined to obtain This completes the proof in this direction.

To prove that is sufficient for the existence of an injective map from to , such a map is exhibited. Set and . Let be a basis of , and be a basis for By Theorem , defines a unique . This is possible, since . Take . By the definition of a basis, can be written uniquely as where . By , , and the linearity of , Since , . Consequently Basis vectors are linearly independent. Hence implies that . Thus . Since is an arbitrary element of , It follows from the first theorem above that is injective. This completes the proof of part (a).