Lagmental Vicfred

Rank--Nullity Splits the Dimension Budget by Vicfred

tags: mathematics, vector spaces, linear maps, dual spaces

The dimension of a finite-dimensional domain is the sum of kernel dimension and image dimension. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Set-up

A linear map \(T:V\to W\) is organized by its kernel \(\ker T\) and image \(\operatorname{im}T\). Quotients and duals express the same information without choosing bases.

$$ T:V\to W $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \dim V=\dim\ker T+\dim\operatorname{im}T $$

The calculation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \left.\begin{aligned}v_1,\ldots,v_k&\text{ basis of }\ker T,\\v_1,\ldots,v_k,u_1,\ldots,u_r&\text{ basis of }V\end{aligned}\right\}\Longrightarrow Tv_1=\cdots=Tv_k=0,\quad Tu_1,\ldots,Tu_r\text{ basis of }\operatorname{im}T $$

What survives abstraction

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad T:V\to W,\\[5pt] \mathsf{C}\;&:\quad \dim V=\dim\ker T+\dim\operatorname{im}T. \end{aligned} $$

The boundary

Dimension formulas below assume finite-dimensional spaces. Infinite-dimensional vector spaces require cardinal arithmetic and may not identify naturally with their double duals.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \dim V=\dim\ker T+\dim\operatorname{im}T \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Sat 28 November 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.