Question

Suppose T : V → W is a homomorphism.

Prove:

(i) If dim(V ) < dim(W) then T is not surjective.

(ii) If dim(V ) > dim(W) then T is not injective.

Answer 1

Theorem : Let V be a finite-dimensional vector space and T : V → W a linear map.

Then range T is a finite-dimensional subspace of W and

dim V = dim null T + dim range T .

(a)

dim null T = dim V − dim range T ≥ dim V − dim W > 0

Since T is injective if and only if dim null T = 0 and we have concluded that dim null T > 0

Threfore , T cannot be injective.

(b)

Also dim range T = dim V − dim null T

≤ dim V

< dim W (given),

Hence the range T cannot be equal to W. So according to the definition of surjective map that range has to be equal to the codomain , T cannot be surjective.