Finding a Basis
for a Subspace

Learn how to identify a minimal spanning set of linearly independent vectors that defines any subspace in $\mathbb{R}^n$.

Linear Algebra

What Is a Subspace?

A subspace $W$ of $\mathbb{R}^n$ is a set of vectors that is closed under addition and scalar multiplication and contains the zero vector $\mathbf{0}$.

✓ Contains $\mathbf{0}$

The zero vector must be in $W$.

✓ Closed Under $+$

If $\mathbf{u}, \mathbf{v} \in W$, then $\mathbf{u}+\mathbf{v} \in W$.

✓ Closed Under $c \cdot$

If $\mathbf{v} \in W$ and $c \in \mathbb{R}$, then $c\mathbf{v} \in W$.

What Is a Basis?

A basis for a subspace $W$ is a set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ that satisfies two conditions simultaneously:

1. Linearly Independent

No vector in the set can be written as a linear combination of the others.

2. Spans $W$

Every vector in $W$ can be expressed as a linear combination of the basis vectors.

Visualizing a Basis in $\mathbb{R}^2$

The Basis Theorem

$\dim(\mathbb{R}^n)$

The dimension of $\mathbb{R}^n$ is $n$. The standard basis has $n$ vectors.

$\dim(\{\mathbf{0}\})$

The zero subspace has dimension $0$. Its basis is the empty set $\emptyset$.

Uniqueness

A basis is not unique, but the number of vectors in any basis always is.

Finding a Basis via Row Reduction

The most common method: place the vectors as rows of a matrix, reduce to row echelon form, and read off the nonzero rows.

Spanning Set vs. Basis vs. Lin. Independent

	Spanning Set	Basis	Lin. Independent
Spans $W$?	✓ Yes	✓ Yes	✕ Maybe not
Lin. Independent?	✕ Maybe not	✓ Yes	✓ Yes
Minimal?	May have extras	✓ Exactly right	May be too few
Size	$\geq \dim(W)$	$= \dim(W)$	$\leq \dim(W)$

How to Find a Basis: Step by Step

1

Form the matrix. Place the given vectors as the rows (for row space) or columns (for column space) of a matrix $A$.

2

Row reduce. Apply Gaussian elimination to bring $A$ to row echelon form (REF) or reduced row echelon form (RREF).

3

Identify pivots. Locate the pivot positions (leading 1s in each row).

4

Read off the basis. For row space: nonzero rows of the echelon form. For column space: the original columns at pivot positions.

5

State the dimension. The number of basis vectors equals $\dim(W)$.

Example 1: Basis for Row Space

Find a basis for the row space of $$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 1 & 1 \end{pmatrix}$$

Example 2: Basis for Column Space

Find a basis for the column space of $$B = \begin{pmatrix} 1 & 3 & 5 \\ 2 & 6 & 10 \\ 1 & 4 & 7 \end{pmatrix}$$

Example 3: Basis for the Null Space

Find a basis for $\text{Nul}(A)$ where $A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \end{pmatrix}$.

Three Fundamental Subspaces

The Standard Basis for $\mathbb{R}^n$

The simplest basis for $\mathbb{R}^n$ is the set of standard unit vectors:

$\mathbb{R}^2$

Basis: $\{\mathbf{e}_1, \mathbf{e}_2\}$. Two vectors, dimension 2.

$\mathbb{R}^3$

Basis: $\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}$. Three vectors, dimension 3.

Which Approach to Use?

Goal	Method	Key Step
Basis for Row Space	Row reduce, take nonzero rows	Use echelon rows
Basis for Column Space	Row reduce, find pivot columns	Use original columns
Basis for Null Space	Solve $A\mathbf{x} = \mathbf{0}$	Parametrize free variables
Basis for Span of vectors	Vectors as rows, row reduce	Discard zero rows

Common Mistakes to Avoid

✕ Using Reduced Columns

For column space, you must use the original columns of $A$ at pivot positions, not the columns of the echelon form.

✕ Forgetting Free Variables

For null space, each free variable contributes one basis vector. Missing a free variable means an incomplete basis.

✕ Redundant Vectors

Including a dependent vector ruins the basis. Always verify linear independence by checking that no row reduces to zero.

Question 1 of 5

A basis for a subspace must satisfy which two properties?

A basis requires exactly linear independence (no redundancy) and spanning (full coverage). Orthogonality and normalization are additional properties of special bases like orthonormal bases, but are not required for a basis in general.

Question 2 of 5

After row reducing a matrix to echelon form, you find 3 nonzero rows and 2 zero rows. What is the dimension of the row space?

The number of nonzero rows in echelon form equals the dimension of the row space. Here there are 3 nonzero rows, so $\dim(\text{Row Space}) = 3$.

Question 3 of 5

To find a basis for the column space of a matrix $A$, after row reducing you should:

For the column space, identify pivot positions in the echelon form, then go back to the original matrix and take those columns. Row operations change column dependencies, so using reduced columns gives the wrong basis.

Question 4 of 5

A $4 \times 6$ matrix $A$ has rank 3. By the Rank-Nullity Theorem, what is $\dim(\text{Nul}(A))$?

The Rank-Nullity Theorem says $\text{rank}(A) + \text{nullity}(A) = n$, where $n$ is the number of columns. So $3 + \text{nullity} = 6$, giving $\text{nullity} = 3$.

Question 5 of 5

Which of the following sets is a basis for $\mathbb{R}^2$?

Option (a) has only 1 vector: it cannot span $\mathbb{R}^2$. Option (b) has two vectors on the same line: they are linearly dependent. Option (c) has 3 vectors in $\mathbb{R}^2$: they cannot all be independent. Option (d) is the standard basis: 2 linearly independent vectors that span $\mathbb{R}^2$.

Finding a Basis
for a Subspace

What Is a Subspace?

What Is a Basis?

Visualizing a Basis in \(\mathbb{R}^2\)

The Basis Theorem

Finding a Basis via Row Reduction

Spanning Set vs. Basis vs. Lin. Independent

How to Find a Basis: Step by Step

Example 1: Basis for Row Space

Example 2: Basis for Column Space

Example 3: Basis for the Null Space

Three Fundamental Subspaces

The Standard Basis for \(\mathbb{R}^n\)

Which Approach to Use?

Common Mistakes to Avoid

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

	Spanning Set	Basis	Lin. Independent
Spans \(W\)?	✓ Yes	✓ Yes	✕ Maybe not
Lin. Independent?	✕ Maybe not	✓ Yes	✓ Yes
Minimal?	May have extras	✓ Exactly right	May be too few
Size	\(\geq \dim(W)\)	\(= \dim(W)\)	\(\leq \dim(W)\)

Finding a Basisfor a Subspace

What Is a Subspace?

What Is a Basis?

Visualizing a Basis in \(\mathbb{R}^2\)

The Basis Theorem

Finding a Basis via Row Reduction

Spanning Set vs. Basis vs. Lin. Independent

How to Find a Basis: Step by Step

Example 1: Basis for Row Space

Example 2: Basis for Column Space

Example 3: Basis for the Null Space

Three Fundamental Subspaces

The Standard Basis for \(\mathbb{R}^n\)

Which Approach to Use?

Common Mistakes to Avoid

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Finding a Basis
for a Subspace