Fundamental Theorem of Linear Algebra — Definition, Formula & Examples

The Fundamental Theorem of Linear Algebra describes how every m×n matrix A gives rise to four subspaces — the column space, row space, null space, and left null space — and specifies their dimensions and orthogonality relationships.

For an m×n matrix $A$ of rank $r$ : the column space $C(A) \subseteq \mathbb{R}^m$ has dimension $r$ ; the row space $C(A^T) \subseteq \mathbb{R}^n$ has dimension $r$ ; the null space $N(A) \subseteq \mathbb{R}^n$ has dimension $n - r$ ; and the left null space $N(A^T) \subseteq \mathbb{R}^m$ has dimension $m - r$ . Furthermore, $C(A) \perp N(A^T)$ and $C(A^T) \perp N(A)$ , meaning each pair consists of orthogonal complements within their respective ambient spaces.

Key Formula

\dim C(A) = \dim C(A^T) = r, \quad \dim N(A) = n - r, \quad \dim N(A^T) = m - r

Where:

$A$ = An m×n matrix
$r$ = Rank of A (number of pivots)
$C(A)$ = Column space of A
$C(A^T)$ = Row space of A
$N(A)$ = Null space of A
$N(A^T)$ = Left null space of A

How It Works

Start by finding the rank

r

of the matrix through row reduction. The number of pivots equals

r

, which is the dimension of both the column space and the row space. The null space then has dimension

n - r

(the number of free variables), and the left null space has dimension

m - r

. The orthogonality part tells you that every vector in the null space is perpendicular to every row of

A

, and every vector in the left null space is perpendicular to every column of

A

. Together, these relationships give a complete geometric picture of what the matrix does.

Worked Example

Problem: Find the dimensions of all four fundamental subspaces for A = [[1, 2, 3], [2, 4, 6]].

Step 1: Row reduce A. Subtract 2 times row 1 from row 2.

A \to \begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \end{bmatrix}

Step 2: Count pivots: there is 1 pivot, so the rank is r = 1. Here m = 2 and n = 3.

r = 1

Step 3: Apply the dimension formulas for each subspace.

\dim C(A) = 1, \quad \dim C(A^T) = 1, \quad \dim N(A) = 3 - 1 = 2, \quad \dim N(A^T) = 2 - 1 = 1

Answer: The column space and row space each have dimension 1, the null space has dimension 2, and the left null space has dimension 1. Note that 1 + 2 = 3 = n and 1 + 1 = 2 = m, confirming the theorem.

Why It Matters

This theorem is the structural backbone of a first course in linear algebra (such as MIT 18.06). It explains why systems of equations have the solution sets they do, underpins least-squares regression in statistics, and clarifies the geometry behind singular value decomposition used throughout data science and engineering.

Common Mistakes

Mistake: Confusing the null space with the left null space, or placing them in the wrong ambient space.

Correction: The null space N(A) lives in R^n (the domain), while the left null space N(A^T) lives in R^m (the codomain). Keep track of which space each subspace belongs to by checking the dimensions of the vectors.