Suppose V and W are both finite-dimensional. Prove that there exists an injective linear map from V to W if and only if dimV≤dimW.
First suppose there exists T∈L(V,W) such that T is injective.
3.16 implies dimnull T=0, then apply 3.22 to get dimV=dimrange T.
Now because range T is a subspace of W it's dimension is less, meaning
dimV=dimrange T≤dimW
This completes the forward direction.
Now suppose dimV≤dimW, Let v1,…,vn be a basis of V and let w1,…,wm be a basis for W. Define Tvj=wj for 1≤j≤n and Tvj=0 for n<j≤m. Clearly null T={0} since each vj is mapped to a nonzero wj, thus 3.16 implies T is injective.