Foundations — Rank-nullity theorem — proof

Before you can even read the theorem $\mathrm{rank}(T)+\mathrm{nullity}(T)=\dim V$, you need to know what each of those words and squiggles means. This page builds them in order, from the plainest idea up to the full statement. Nothing here is assumed — if the parent note used it, we define it here.

0. What is a vector? (the atom)

Why do we start here? Because everything — spaces, maps, kernels — is built from arrows. If you cannot picture an arrow, no later symbol will feel real.

The two numbers are the vector's components. Adding two vectors means placing them tip-to-tail; scaling a vector by a number $c$ stretches it (or flips it, if $c<0$).

1. What is a vector space $V$? (a room full of arrows)

The letters $V$ and $W$ in the theorem are just names for two such rooms — the input room and the output room.

Vectors don't have to be arrows! Polynomials like $2x^2+x-5$ also add and scale like arrows, so the set of all polynomials of degree $\le 3$ is a vector space too (the parent's Example 3). That is why the theorem works for "abstract" spaces. See Dimension of a Vector Space for the full story.

2. The field $F$ (which numbers we scale by)

You mostly ignore $F$ — read it as "the ordinary numbers" — but it must exist so that "scale by $c$" has meaning.

3. Linear independence & span (do arrows overlap, or fill space?)

Two ideas that the whole proof leans on.

4. Basis and dimension (the minimal complete kit)

The symbol $n$ in the proof is just $\dim V$ given a short name. "$V$ finite-dimensional" means $n$ is a finite number — you can list a basis and stop.

The move "extend a basis of a subspace to a basis of the whole space" is the Basis Extension Theorem, the engine of Step 1.

5. Linear map $T:V\to W$ (the sorting machine)

The picture: $T$ takes the grid of $V$ and stretches, rotates, or flattens it into $W$ — but keeps grid lines straight and evenly spaced (that's what linearity looks like). Some maps flatten a whole direction onto zero — that flattening is the kernel, next.

6. The zero vector $0$ and the kernel $\ker T$

The kernel is a subspace of the input room $V$. More on it in Kernel and Image of a Linear Map.

7. The image $\mathrm{im}T$ and rank

The image is a subspace of the output room $W$. Crucial subtlety the parent hammers: rank + nullity sums to $\dim V$ (the domain), not $\dim W$ — because we are counting how the input directions get sorted.

8. Matrices as linear maps

A matrix like $A=\left(\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\end{array}\right)$ is a linear map: feeding it a column vector $x$ and computing $Ax$ takes a vector from ${\mathbb{R}}^3$ to ${\mathbb{R}}^2$. Solving $Ax=0$ finds the kernel; the number of free variables in that solution is the nullity — that link is Solving Linear Systems — free variables count. When a map loses no directions (kernel is just $\{0\}$) it is an isomorphism, see Invertible Linear Maps and Isomorphisms.

How the pieces feed the theorem

Read top to bottom: arrows are atoms, arrows build spaces and independence, independence builds bases, bases give dimension, and dimension of kernel + image is exactly what the theorem measures.

Equipment checklist

Test yourself — cover the right side and answer out loud.