Kernel-Image Theorem

up:: Linear Transformation

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

Given a linear transformation $T:U\to V$ whose domain is the Vector Space $(U,+,\cdot )$, we have that

$$\dim U=\dim (ImT)+\dim (\ker T)$$

Proof

Assume that $\dim U=n$, and that $\ker T\ne \{0\}$ at first.

Then $\ker T$ will have a Hamel Basis $K=\{k_1,\dots ,k_m\}$¹.
Since Any linearly independent set can be extended to a Hamel basis, we can extend it to a basis of $U$ itself by uniting with vectors $S=\{s_{m+1},\dots ,s_n\}$ outside of its span.
Thus, $B\equiv K\cup S$ is a basis for $U$.

Since A linear transformation’s image of a basis spans its image, we have that $T(B)$ is a Spanning Set for $ImT$ ─ but note that²

$$T(B)=T(S)=\{T(s_{m+1}),\dots ,T(s_n))\}$$

Thus, $ImT$ is spanned by the $n-m$ vectors in $T(S)$. We seek to prove that it is a basis ─ thus, that it is also linearly independent. Thus, we seek to prove that ${\sum }_k{\alpha }_{k}T(s_k)=0⟺{\alpha }_k=0$.

Consider the Linear Combination

$$\sum _{k=m+1}^n{\alpha }_{k}T(s_k)=T\left(\sum _{k=m+1}^n{\alpha }_k{s}_k\right)$$

which implies that ${\sum }_{k=m+1}^n{\alpha }_k{s}_k\in \ker T$ ─ thus, it can be written as a linear combination of its basis $K$:

$$\sum _{k=m+1}^n{\alpha }_k{s}_k=\sum _{i=1}^m{\alpha }_i{k}_i⟺\sum _{i=1}^m{\tilde{\alpha }}_i{k}_i+\sum _{k=m+1}^n{\tilde{\alpha }}_k{s}_k=0$$

where we invoke that $B=K\cup S$ is linearly independent, for which this sum only equals $0$ if all coefficients are also $0$.

Thus, $T(S)$ is a basis for $ImT$, and we can conclude that

$$\dim U=\dim \ker T+\dim ImT$$

If $\ker T=\{0\}$, we need only consider a basis for $U$, 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. ↩