Rank-Nullity and Conditions for Invertibility

The Key Theorem of Linear Algebra is not a single equation, it is a tightly woven web of equivalences that ties together every concept in the chapter. Rank, nullity, determinant, invertibility, kernel, image, columns-as-basis, unique solvability: for square matrices, these are all the same condition, just dressed in different language. Mastering one of them unlocks all the others. This is why an exam question that looks like it requires a long calculation often dissolves into a one-line answer once you spot which piece of the web is being tested.

The Rank-Nullity Theorem

For any $m \times n$ matrix $A$, or equivalently any linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$:

$\dim \ker(A) + \dim \operatorname{im}(A) = n$

Here $\operatorname{rank}(A) = \dim \operatorname{im}(A)$ counts the pivot columns after row-reduction, and $\operatorname{nullity}(A) = \dim \ker(A)$ counts the free variables. They must sum to the number of input columns. This is the geometric content of "what goes in, must come out either as image or as kernel collapse".

The web of equivalences for square matrices

For an $n \times n$ matrix $A$, the following statements are all equivalent:

• $A$ is invertible.
• $\det A \neq 0$.
• $\operatorname{rank}(A) = n$ (full rank).
• $\ker A = \{\mathbf{0}\}$ (trivial kernel).
• The columns of $A$ form a basis for $\mathbb{R}^n$.
• $A \mathbf{x} = \mathbf{b}$ has a unique solution for every $\mathbf{b}$.
• $0$ is not an eigenvalue of $A$.
• The reduced row echelon form of $A$ is the identity matrix $I_n$