Suppose V and W are both finite-dimensional. Prove that there exists a surjective linear map from V onto W if and only if dimV≥dimW.
First suppose T∈L(V,W) is surjective, meaning range T=W, combine this with 3.22 to get
dimV=dimrange T+dimnull T≤dimrange T=dimW
Which completes the forward direction.
Now suppose dimV≥dimW, let v1,…,vn be a basis for V and w1,…,wm be a basis for W. Define Tvj=wj for 1≤j≤m and define Tvj=0 for m<j≤n.
T is clearly surjective as
range T=span(Tv1,…,Tvn)=span(w1,…,wm)=W
which completes the proof.