Rank-Nullity Theorem

The Theorem

For a linear transformation $L: V \rightarrow W$ between finite-dimensional vector spaces:

Key Implications

1. Injectivity: $L$ is injective if and only if $\dim(\text{Ker}(L)) = 0$

2. Surjectivity: $L$ is surjective if and only if $\dim(\text{Im}(L)) = \dim(W)$

3. Dimension Constraint: If $\dim(V) < \dim(W)$, then $L$ cannot be surjective

4. Dimension Constraint: If $\dim(V) > \dim(W)$, then $L$ cannot be injective

Example

For $L: \mathbb{R}^4 \rightarrow \mathbb{R}^3$ with $\dim(\text{Ker}(L)) = 1$:

• $\dim(\text{Im}(L)) = \dim(V) - \dim(\text{Ker}(L)) = 4 - 1 = 3$
• Since $\dim(\text{Im}(L)) = \dim(W) = 3$, $L$ is surjective
• Since $\dim(\text{Ker}(L)) \neq 0$, $L$ is not injective