Relationship Between Rank, Nullity, Determinant and Invertibility

Theorem

Let V,W be vector spaces over F with dimV=dimW finite and T:VW linear.

The following are all equivalent:

  1. T is invertible (bijective)
  2. nullity(T)=0
  3. rank(T)=dimW
  4. det(T)0

Using the same notation in the theorem above, we prove this using the following lemmas.

Lemma

nullity(T)=0 if and only if T is injective.

Proof

Assume that nullity(T)=0 and let a,bV such that T(a)=T(b). This implies that

0=T(a)T(b)=T(ab).

Then since nullity(T)=0, the set of values ab which map to zero contains only the zero dimensional subspace {0}, and hence ab=0 which implies a=b.

On the converse, if T is assumed to be injective then

T(a)=T(b)a=b

which is equivalent to

T(ab)=0ab=0.

With ab an arbitrary vector by setting b=0, we have that the only thing that maps to zero is the zero vector itself, and therefore the subspace of elements mapped to zero is zero dimensional.

Lemma

rank(T)=dimV if and only if T is surjective.

Proof

If rank(T)=dimW then T must be surjective, as an n dimensional subspace of an n dimensional space must be the whole space.

Similarly if T is surjective, then T(x) is any arbitrary element of W, and hence T is full rank.


Proof

We can now prove the main result. (2) and (3) are equivalent from the rank nullity theorem. Since (2) implies injectivity, and (3) implies surjectivity, but (2) is equivalent to (3), each of these independently prove bijectivity, (1). The converse is also true, with (1) implying T is both surjective and injective, hence proving both (2) and (3).

Hence we have established (1)(2)(3).

Then from matrix has non-zero determinant if and only if it is invertible we can conclude the final equivalenc.