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Column Space
& Null Space

Discover the two fundamental subspaces of any matrix: where \(A\mathbf{x}\) can reach, and where \(A\mathbf{x} = \mathbf{0}\) hides.

Linear Algebra

What Does a Matrix Really Do?

A matrix \(A\) is a function. When you multiply \(A\mathbf{x}\), you transform the input vector \(\mathbf{x}\) into an output vector \(\mathbf{b}\). The natural question is: which outputs are possible, and which inputs produce zero?

The Column Space \(\text{Col}(A)\)

The column space of a matrix \(A\) is the set of all possible linear combinations of its columns. If \(A\) has columns \(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\), then:

Equivalently, \(\text{Col}(A)\) is the set of all vectors \(\mathbf{b}\) such that \(A\mathbf{x} = \mathbf{b}\) has a solution. It is a subspace of \(\mathbb{R}^m\) (the output space).

Visualizing Column Space

\(\mathbf{a}_1\) \(\mathbf{a}_2\) \(\text{Col}(A)\) x z y O

The Null Space \(\text{Nul}(A)\)

The null space of a matrix \(A\) is the set of all vectors \(\mathbf{x}\) that get mapped to the zero vector:

The null space is a subspace of \(\mathbb{R}^n\) (the input space). It always contains the zero vector, and it captures the "hidden directions" that the matrix collapses to nothing.

Visualizing Null Space

Input Space \(\mathbb{R}^n\) \(\text{Nul}(A)\) \(A\) Output Space \(\mathbb{R}^m\) \(\text{Col}(A)\) \(\mathbf{0}\)

The Rank-Nullity Theorem

The number of columns is split between independent directions (rank) and collapsed directions (nullity). They always add up to the total number of columns \(n\).

Rank
Number of pivot columns after row reduction. Dimension of \(\text{Col}(A)\).
Nullity
Number of free variables. Dimension of \(\text{Nul}(A)\).

Rank + Nullity = \(n\)

Rank Nullity n columns total

Col(A) vs Nul(A): Key Differences

Column Space Null Space
Definition \(\{A\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n\}\) \(\{\mathbf{x} \mid A\mathbf{x}=\mathbf{0}\}\)
Lives in \(\mathbb{R}^m\) (output) \(\mathbb{R}^n\) (input)
Dimension rank(\(A\)) nullity(\(A\))
Found by Pivot columns of \(A\) Solving \(A\mathbf{x}=\mathbf{0}\)
Question answered What can \(A\) produce? What does \(A\) kill?

How to Find the Column Space

1
Row reduce the matrix \(A\) to echelon form (REF or RREF).
2
Identify pivot columns: the columns containing leading 1s (pivots).
3
Go back to the original matrix \(A\). Take the corresponding columns from \(A\) (not the reduced form).
4
Those original columns form a basis for \(\text{Col}(A)\). Their count is the rank.

How to Find the Null Space

1
Set up the augmented matrix \([A \mid \mathbf{0}]\).
2
Row reduce to RREF.
3
Identify free variables: columns without pivots. Set each free variable to a parameter (\(t, s, \ldots\)).
4
Express pivot variables in terms of free variables. Write the general solution as a linear combination.
5
The vectors multiplied by the parameters form a basis for \(\text{Nul}(A)\).

Example 1: Find Col(A) and Nul(A)

Step 1: Row reduce \(A\):

Pivots in columns 1 and 3. Column 2 is free. Rank = 2, Nullity = 1.

Example 1: Results

Basis for Col(A)
Pivot columns from original \(A\):
\(\left\{\begin{pmatrix}1\\2\\3\end{pmatrix},\;\begin{pmatrix}1\\3\\4\end{pmatrix}\right\}\)
Basis for Nul(A)
Solve \(A\mathbf{x}=\mathbf{0}\). Free var \(x_2 = t\):
\(\left\{\begin{pmatrix}-2\\1\\0\end{pmatrix}\right\}\)

Example 2: A Wider Matrix

Row reduce:

Only 1 pivot (column 1). Free variables: \(x_2, x_3, x_4\). Rank = 1, Nullity = 3.

Example 3: Invertible Matrix

Col(A)
Both columns are pivots. \(\text{Col}(A) = \mathbb{R}^2\): the entire output space. Every \(\mathbf{b}\) is reachable.
Nul(A)
No free variables. The only solution to \(A\mathbf{x}=\mathbf{0}\) is \(\mathbf{x}=\mathbf{0}\). Null space = \(\{\mathbf{0}\}\).

Common Mistakes to Avoid

✕ Using RREF columns as basis
The column space basis comes from the original matrix columns, not the reduced form. RREF only tells you which columns are pivots.
✕ Confusing the spaces' homes
\(\text{Col}(A)\) lives in \(\mathbb{R}^m\) (output). \(\text{Nul}(A)\) lives in \(\mathbb{R}^n\) (input). They are in different spaces (unless \(m=n\)).
✕ Forgetting the zero vector
Both subspaces must contain \(\mathbf{0}\). If your proposed basis doesn't span a set including \(\mathbf{0}\), something is wrong.

Question 1 of 5

The column space of an \(m \times n\) matrix \(A\) is a subspace of which space?

Question 2 of 5

If a \(4 \times 6\) matrix has rank 3, what is the dimension of its null space?

Question 3 of 5

When finding a basis for \(\text{Col}(A)\), you should take the pivot columns from:

Question 4 of 5

If \(A\) is an invertible \(n \times n\) matrix, what is \(\text{Nul}(A)\)?

Question 5 of 5

Consider \(A = \begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix}\). Which vector is in \(\text{Nul}(A)\)?