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Injectivity and Nullity

The precise algebraic characterization of injectivity: $A$ is injective if and only if $\text{Nullity}(A) = 0$ . Equivalently, $A\mathbf{x} = \mathbf{0}$ has only the trivial solution.

Why? If $A\mathbf{x}_1 = A\mathbf{x}_2$ , then $A(\mathbf{x}_1 - \mathbf{x}_2) = \mathbf{0}$ , so $\mathbf{x}_1 - \mathbf{x}_2 \in \text{Null}(A)$ . If the null space is trivial, then $\mathbf{x}_1 - \mathbf{x}_2 = \mathbf{0}$ , so $\mathbf{x}_1 = \mathbf{x}_2$ .

This gives a clean test: to check if $A$ is injective, solve $A\mathbf{x} = \mathbf{0}$ and check if the only solution is $\mathbf{x} = \mathbf{0}$ .

Formal View

Theorem 3.9

A \in \mathbb{R}^{m \times n}

is injective

\Longleftrightarrow

\text{Nullity}(A) = 0

\Longleftrightarrow

The equation

A\mathbf{x} = \mathbf{0}

has only the trivial solution

\mathbf{x} = \mathbf{0}

Why This Matters

The nullity gives a computable measure of injectivity failure.

Nullity = 0 means the system $A\mathbf{x} = \mathbf{b}$ has at most one solution
In machine learning, large nullity in a feature matrix signals collinearity issues

Learning Resources

Null space relates to injective maps

MIT OpenCourseWare

Lecture 10: Null space and injectivity.

49 min

Quiz

Question 1

If $A$ has nullity 0, then $A\mathbf{x} = \mathbf{b}$ has at most one solution for any $\mathbf{b}$ .

Common Mistakes

Confusing "at most one solution" (injectivity) with "exactly one solution" (injectivity + surjectivity = bijectivity).