Kernel-Image Theorem

up:: Linear Transformation

The Dimension of a Vector Space is equal to the dimension of ‘s Kernel plus its Linear Transformation Rank.

Given a linear transformation whose domain is the Vector Space , we have that

Proof

Assume that , and that at first.

Then will have a Hamel Basis ¹.
Since Any linearly independent set can be extended to a Hamel basis, we can extend it to a basis of itself by uniting with vectors outside of its span.
Thus, is a basis for .

Since A linear transformation’s image of a basis spans its image, we have that is a Spanning Set for ─ but note that²

Thus, is spanned by the vectors in . We seek to prove that it is a basis ─ thus, that it is also linearly independent. Thus, we seek to prove that .

Consider the Linear Combination

which implies that ─ thus, it can be written as a linear combination of its basis :

where we invoke that is linearly independent, for which this sum only equals if all coefficients are also .

Thus, is a basis for , and we can conclude that

If , we need only consider a basis for , for which we’ll reach the same conclusion.

Corollaries

• An endomorphism is injective iff it’s surjective

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References

• Um Curso de Álgebra Linear, Flávio Ulhoa Coelho & Mary Lilian Lourenço. Editora EDUSP.

Footnotes

1. Since it will have a linearly independent set, and since Every vector space which has a linearly independent set has a Hamel Basis containing this set. ↩

2. Invokes that The image of the union is the union of the images. ↩