Suppose T is a linear map from F4 to F2 such that
null T={(x1,x2,x3,x4)∈F4:x1=5x2 and x3=7x4}
Prove that T is surjective.
By 3.22
dimrange T+dimnull T=4
Since dimnull T=2 (exercise to the reader) this implies dimrange T=2 so we can let v1,v2 be a basis of range T, extending (2.33) this basis to a basis of F2 doesn't add anything, implying v1,v2 is already a basis for F2.
I feel there should be a theoremin the book I could cite to show if U is a subspace of V, U=V iff dimU=dimV.
Proof: Let u1,…,um be a basis of U and extend it to a basis of V. Since dimU=dimV extending it does nothing, thus u1,…,um is already a basis for V and so U=V.