6: Subspaces

Subspaces are the building blocks of linear algebra,they’re the sets where vector operations stay contained, and understanding them reveals the geometry hidden inside matrices.

Basic Definitions

(Subspace)

A subset $S \subseteq \mathbb{R}^n$ is a subspace of $\mathbb{R}^n$ if and only if it satisfies three conditions:

1. Contains the zero vector: $\mathbf{0} \in S$
2. Closed under addition: If $u, v \in S$, then $u + v \in S$
3. Closed under scalar multiplication: If $v \in S$ and $c \in \mathbb{R}$, then $cv \in S$

In other words, a subspace is a set that “behaves like a vector space” within $\mathbb{R}^n$.

(Why These Three Conditions?)

The intuition: a subspace must be self-contained under the operations that define vector spaces.

• The zero vector ensures you have a natural “origin”
• Closure under addition means you can combine vectors freely
• Closure under scalar multiplication means you can scale vectors freely

Together, these guarantee that all linear combinations of vectors in $S$ stay inside $S$.

Examples

Example 1: The Trivial Subspaces

For any $n$:

• $S = \{\mathbf{0}\}$ is always a subspace (the trivial subspace)
• $S = \mathbb{R}^n$ is always a subspace (the whole space)

Example 2: Lines Through the Origin

The set

where $v \neq \mathbf{0}$ is a subspace of $\mathbb{R}^n$.

This is the line through the origin in the direction of $v$.

Why?

• Contains $\mathbf{0}$: Set $t = 0$
• Closed under addition: $t_1 v + t_2 v = (t_1 + t_2)v \in S$
• Closed under scalar multiplication: $c(tv) = (ct)v \in S$

Example 3: Planes Through the Origin

The set

where $v_1, v_2$ are linearly independent is a subspace.

This is the plane spanned by $v_1$ and $v_2$.

Example 4: Not a Subspace

The set

is not a subspace because:

• It doesn’t contain $\mathbf{0}$ (since $0 + 0 \neq 1$)
• It’s not closed under addition: $(1, 0)$ and $(0, 1)$ are in $S$, but their sum $(1, 1)$ is not

This is a line, but it doesn’t. pass through the origin,so it fails to be a subspace.

Key Result: Span is Always a Subspace

Theorem: For any collection of vectors $v_1, \dots, v_k \in \mathbb{R}^n$, the span

is a subspace of $\mathbb{R}^n$.

Proof:

1. Contains $\mathbf{0}$: Take all coefficients $c_i = 0$, then $0v_1 + \cdots + 0v_k = \mathbf{0}$.

2. Closed under addition: If $u = c_1 v_1 + \cdots + c_k v_k$ and $w = d_1 v_1 + \cdots + d_k v_k$, then

   $$u + w = (c_1 + d_1)v_1 + \cdots + (c_k + d_k)v_k \in \operatorname{span}\{v_1, \dots, v_k\}.$$

3. Closed under scalar multiplication: If $u = c_1 v_1 + \cdots + c_k v_k$ and $a \in \mathbb{R}$, then

   $$au = (ac_1)v_1 + \cdots + (ac_k)v_k \in \operatorname{span}\{v_1, \dots, v_k\}.$$

This is why span is so important,it’s the canonical way to construct subspaces.

Column Space and Null Space

Two fundamental subspaces arise from any matrix $A \in \mathbb{R}^{m \times n}$:

(Column Space)

The column space of $A$, denoted $\operatorname{col}(A)$, is the span of the columns of $A$:

This is a subspace of $\mathbb{R}^m$.

Interpretation: $\operatorname{col}(A)$ is the set of all possible outputs $Ax$ as $x$ ranges over $\mathbb{R}^n$.

(Null Space)

The null space (or kernel) of $A$, denoted $\operatorname{null}(A)$ or $\ker(A)$, is the set of all solutions to $Ax = \mathbf{0}$:

This is a subspace of $\mathbb{R}^n$.

Why is it a subspace?

1. $A\mathbf{0} = \mathbf{0}$, so $\mathbf{0} \in \operatorname{null}(A)$
2. If $Au = \mathbf{0}$ and $Av = \mathbf{0}$, then $A(u + v) = Au + Av = \mathbf{0}$
3. If $Au = \mathbf{0}$, then $A(cu) = c(Au) = c\mathbf{0} = \mathbf{0}$

Interpretation: The null space captures all the “redundancy” in the matrix,the directions that get collapsed to zero.

Dimension and Basis

(Basis)

A basis for a subspace $S$ is a linearly independent set of vectors whose span equals $S$.

Equivalently, a basis is a minimal spanning set or a maximal independent set.

(Dimension)

The dimension of a subspace $S$, denoted $\dim(S)$, is the number of vectors in any basis for $S$.

Key fact: All bases for a given subspace have the same size.

Examples

• $\dim(\{\mathbf{0}\}) = 0$
• $\dim(\mathbb{R}^n) = n$
• A line through the origin has dimension $1$
• A plane through the origin has dimension $2$

The Rank-Nullity Theorem

For any $m \times n$ matrix $A$:

Or equivalently:

Interpretation: The dimension of the input space ($n$) is split between:

• The dimension of directions that get mapped somewhere non-trivial (the rank)
• The dimension of directions that collapse to zero (the nullity)

This is one of the most important equations in linear algebra.

Computing Subspaces from RREF

Given a matrix $A$, we can compute both $\operatorname{col}(A)$ and $\operatorname{null}(A)$ from $\operatorname{rref}(A)$.

Finding a Basis for $\operatorname{col}(A)$

1. Compute $\operatorname{rref}(A)$
2. Identify the pivot columns in $\operatorname{rref}(A)$
3. The corresponding columns from the original matrix $A$ form a basis for $\operatorname{col}(A)$

Why the original columns? Because row operations preserve the column space relationships but change the actual vectors. The pivot positions tell you which original columns are independent.

Finding a Basis for $\operatorname{null}(A)$

1. Compute $\operatorname{rref}(A)$
2. Identify the free variables (columns without pivots)
3. For each free variable, set it to $1$ and all other free variables to $0$, then solve for the basic variables
4. Each solution vector forms a basis vector for $\operatorname{null}(A)$

The dimension of $\operatorname{null}(A)$ equals the number of free variables.

Why Subspaces Matter

Subspaces are the geometric objects that linear algebra studies.

• Solutions to linear systems live in subspaces
• Eigenvectors span eigenspaces (which are subspaces)
• Projections map onto subspaces
• Least-squares problems minimize distance to subspaces

Understanding subspaces means understanding the structure of $\mathbb{R}^n$, not just individual vectors.