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Row, Column &
Null Space

Explore the fundamental subspaces of a matrix and discover how row space, column space, and null space relate to each other through rank and dimension.

Linear Algebra

What Is a Subspace?

A subspace of \(\mathbb{R}^n\) is a set of vectors that is closed under addition and scalar multiplication, and always contains the zero vector. Every matrix \(A\) defines several important subspaces.

Closure
If \(\mathbf{u}\) and \(\mathbf{v}\) are in the set, so is \(\mathbf{u}+\mathbf{v}\) and \(c\mathbf{u}\).
Zero Vector
The set must contain \(\mathbf{0}\). If it doesn't, it cannot be a subspace.

The Three Fundamental Subspaces

Given an \(m \times n\) matrix \(A\), three subspaces capture everything about how \(A\) transforms vectors:

Column Space
\(\text{Col}(A)\): all vectors \(\mathbf{b}\) such that \(A\mathbf{x}=\mathbf{b}\) has a solution. Lives in \(\mathbb{R}^m\).
Row Space
\(\text{Row}(A)\): span of the rows of \(A\). Equivalently, \(\text{Col}(A^T)\). Lives in \(\mathbb{R}^n\).
Null Space
\(\text{Null}(A)\): all \(\mathbf{x}\) with \(A\mathbf{x}=\mathbf{0}\). The "kernel" of the transformation. Lives in \(\mathbb{R}^n\).

Visualizing the Map

\(\mathbb{R}^n\)
\(\text{Row}(A)\)
\(\text{Null}(A)\)
\(A\)
\(\mathbb{R}^m\)
\(\text{Col}(A)\)
\(\mathbf{0}\)
\(\text{Null}(A)\) maps to \(\mathbf{0}\)

Column Space: Where Can We Reach?

The column space of \(A\) is the span of its column vectors. It answers: for which \(\mathbf{b}\) does \(A\mathbf{x} = \mathbf{b}\) have a solution?

Finding a basis
Row reduce \(A\). The original columns in pivot positions form a basis for \(\text{Col}(A)\).
Dimension
\(\dim(\text{Col}(A)) = \text{rank}(A)\), the number of pivot columns.

Row Space: Constraints on the Domain

The row space is the span of the row vectors. Row reduction preserves the row space, so the nonzero rows of any echelon form give a basis.

1
Row reduce \(A\) to echelon form \(R\).
2
Read off the nonzero rows of \(R\): they form a basis for \(\text{Row}(A)\).
3
\(\dim(\text{Row}(A)) = \text{rank}(A)\), the same as the column space dimension.

Null Space: What Gets Crushed?

The null space contains every vector that \(A\) sends to the zero vector. It measures the "redundancy" or "degrees of freedom" in the system.

Finding a basis
Row reduce \(A\) and solve \(A\mathbf{x}=\mathbf{0}\). Each free variable produces one basis vector.
Dimension
\(\dim(\text{Null}(A)) = n - \text{rank}(A)\), the nullity.

The Rank-Nullity Theorem

This is also called the Dimension Theorem. It says the number of columns is split between pivot columns (rank) and free columns (nullity).

n columns
\(\text{rank}(A) = r\)
\(\text{nullity} = n - r\)
 + pivot columns + free columns = total columns

A Surprising Equality

One of the most elegant results in linear algebra: the row space and column space always have the same dimension, even though they live in different spaces.

Row Space
Lives in \(\mathbb{R}^n\) (domain). Dimension = \(r\).
Column Space
Lives in \(\mathbb{R}^m\) (codomain). Dimension = \(r\).
Null Space
Lives in \(\mathbb{R}^n\) (domain). Dimension = \(n - r\).

Row Space \(\perp\) Null Space

The row space and null space are orthogonal complements in \(\mathbb{R}^n\): every vector in the domain can be uniquely decomposed into a row-space part and a null-space part.

Row(A) Null(A) x Every x = (row part) + (null part)

Column Space vs. Row Space

Column Space Row Space
Definition Span of columns of \(A\) Span of rows of \(A\)
Lives in \(\mathbb{R}^m\) \(\mathbb{R}^n\)
Dimension \(\text{rank}(A)\) \(\text{rank}(A)\)
Basis from Pivot columns of original \(A\) Nonzero rows of echelon form
Orthogonal to \(\text{Null}(A^T)\) (left null space) \(\text{Null}(A)\)

Step-by-Step: Find All Three

1
Row reduce \(A\) to reduced row echelon form (RREF).
2
Identify pivots. Count \(r\) pivot positions: that is your rank.
3
Column space basis: take the original columns of \(A\) in pivot positions.
4
Row space basis: take the nonzero rows of the RREF.
5
Null space basis: solve \(A\mathbf{x}=\mathbf{0}\). Each free variable gives one basis vector. Nullity \(= n - r\).

Example 1: A \(2 \times 3\) Matrix

1
RREF: \(R_2 \leftarrow R_2 - 2R_1\) gives \(\begin{pmatrix} 1 & 2 & 1 \\ 0 & 0 & 1 \end{pmatrix}\). Pivots in columns 1 and 3.
2
Rank \(= 2\). Nullity \(= 3 - 2 = 1\).
3
Col(A) basis: original columns 1 and 3: \(\left\{\begin{pmatrix}1\\2\end{pmatrix},\;\begin{pmatrix}1\\3\end{pmatrix}\right\}\).
4
Row(A) basis: nonzero RREF rows: \(\{(1,2,1),\;(0,0,1)\}\). Equivalently \(\{(1,2,0),\;(0,0,1)\}\) after further reduction.
5
Null(A) basis: free variable \(x_2 = t\). Solution: \(\mathbf{x} = t\begin{pmatrix}-2\\1\\0\end{pmatrix}\). Basis: \(\left\{\begin{pmatrix}-2\\1\\0\end{pmatrix}\right\}\).

Example 2: A \(3 \times 3\) Matrix

RREF: \(R_3 \leftarrow R_3 - R_1 - R_2\) yields \(\begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{pmatrix}\). Pivots in columns 1 and 2. Rank \(= 2\), nullity \(= 1\).

Col(A)
Basis: \(\left\{\begin{pmatrix}1\\0\\1\end{pmatrix},\;\begin{pmatrix}0\\1\\1\end{pmatrix}\right\}\) (original columns 1, 2).
Row(A)
Basis: \(\{(1,0,2),\;(0,1,-1)\}\) from RREF rows.
Null(A)
Basis: \(\left\{\begin{pmatrix}-2\\1\\1\end{pmatrix}\right\}\). Check: \(A\begin{pmatrix}-2\\1\\1\end{pmatrix} = \mathbf{0}\) ✓

Example 3: Full Rank Matrix

Since \(A\) is invertible, its RREF is \(I_2\). Every column is a pivot column.

Col(A)
\(= \mathbb{R}^2\) (all of the codomain). Rank \(= 2\).
Row(A)
\(= \mathbb{R}^2\) (all of the domain). Rank \(= 2\).
Null(A)
\(= \{\mathbf{0}\}\) only. Nullity \(= 0\). The only solution to \(A\mathbf{x}=\mathbf{0}\) is \(\mathbf{x}=\mathbf{0}\).

Common Mistakes to Avoid

✕ Using RREF columns for Col(A)
The column space basis comes from the original matrix columns, not the reduced columns. Row reduction changes column relationships.
✕ Confusing the spaces' homes
Row space and null space live in \(\mathbb{R}^n\). Column space lives in \(\mathbb{R}^m\). Mixing them up leads to dimension mismatches.
✕ Forgetting the zero vector
Writing null space as \(\{(-2, 1, 0)\}\) instead of \(\text{span}\{(-2, 1, 0)\}\). The null space is a subspace, not a single vector.

Summary of Relationships

Col(A) Row(A) Null(A)
Lives in \(\mathbb{R}^m\) \(\mathbb{R}^n\) \(\mathbb{R}^n\)
Dimension \(r\) \(r\) \(n - r\)
Basis from Pivot cols of \(A\) Nonzero rows of RREF Free variables of \(A\mathbf{x}=\mathbf{0}\)
Orthogonal to \(\text{Null}(A^T)\) \(\text{Null}(A)\) \(\text{Row}(A)\)
Key identity \(\text{rank}(A) + \text{nullity}(A) = n\)

Question 1 of 5

If \(A\) is a \(4 \times 6\) matrix with \(\text{rank}(A) = 3\), what is \(\dim(\text{Null}(A))\)?

Question 2 of 5

To find a basis for the column space of \(A\), you should take:

Question 3 of 5

The row space and null space of a matrix are related by:

Question 4 of 5

If a \(3 \times 3\) matrix \(A\) is invertible, what is its null space?

Question 5 of 5

For any matrix \(A\), which two subspaces always have the same dimension?