# Chapter 18. Subspaces

# Chapter 18. Subspaces

A subspace is a subset of a vector space that is itself a vector space under the same operations. Subspaces are among the most important objects in linear algebra because many problems reduce to understanding smaller vector spaces inside larger ones.

Lines through the origin, planes through the origin, solution spaces of homogeneous systems, null spaces, column spaces, and eigenspaces are all examples of subspaces.

The concept of subspace formalizes the idea of a linear part of a vector space.

## 18.1 Definition of a Subspace

Let \(V\) be a vector space over a field \(F\). A subset \(W \subseteq V\) is called a subspace of \(V\) if \(W\) is itself a vector space under the same addition and scalar multiplication operations as \(V\).

The operations are inherited from the ambient space. No new operations are introduced.

For \(W\) to be a subspace, every vector operation performed inside \(W\) must remain inside \(W\).

## 18.2 Subspace Test

The vector space axioms do not need to be checked one by one. A simpler criterion exists.

A nonempty subset \(W \subseteq V\) is a subspace if and only if:

1. \(u + v \in W\) whenever \(u, v \in W\),
2. \(cv \in W\) whenever \(c \in F\) and \(v \in W\).

These conditions are called closure under addition and closure under scalar multiplication.

Since the subset already lies inside a vector space, the remaining vector space axioms are inherited automatically.

The condition that \(W\) is nonempty is essential. Without it, the empty set would satisfy the closure conditions vacuously.

## 18.3 The Zero Vector

Every subspace contains the zero vector.

To see this, let \(v \in W\). Since \(W\) is closed under scalar multiplication,

$$
0v \in W.
$$

But

$$
0v = 0.
$$

Therefore

$$
0 \in W.
$$

This gives a fast test for rejecting candidate subspaces. If a set does not contain the zero vector, then it cannot be a subspace.

For example,

$$
S = \{(x,y) \in \mathbb{R}^2 : x + y = 1\}
$$

is not a subspace because

$$
(0,0) \notin S.
$$

## 18.4 Geometric Interpretation

In \(\mathbb{R}^2\), the subspaces are:

| Subspace | Dimension |
|---|---|
| \(\{0\}\) | 0 |
| Any line through the origin | 1 |
| \(\mathbb{R}^2\) | 2 |

In \(\mathbb{R}^3\), the subspaces are:

| Subspace | Dimension |
|---|---|
| \(\{0\}\) | 0 |
| Lines through the origin | 1 |
| Planes through the origin | 2 |
| \(\mathbb{R}^3\) | 3 |

The phrase “through the origin” is critical. A translated line or plane is generally not a subspace because it does not contain the zero vector.

For example,

$$
\{(x,y) : y = 2x\}
$$

is a subspace of \(\mathbb{R}^2\), but

$$
\{(x,y) : y = 2x + 1\}
$$

is not.

## 18.5 Examples of Subspaces

Consider the subset

$$
W =
\left\{
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
\in \mathbb{R}^3
:
x + y + z = 0
\right\}.
$$

We verify the subspace conditions.

First, the zero vector belongs to \(W\) because

$$
0 + 0 + 0 = 0.
$$

Next, let

$$
u =
\begin{bmatrix}
x_1 \\
y_1 \\
z_1
\end{bmatrix},
\qquad
v =
\begin{bmatrix}
x_2 \\
y_2 \\
z_2
\end{bmatrix}
$$

be vectors in \(W\). Then

$$
x_1 + y_1 + z_1 = 0
$$

and

$$
x_2 + y_2 + z_2 = 0.
$$

Their sum satisfies

$$
(x_1+x_2) + (y_1+y_2) + (z_1+z_2) =
(x_1+y_1+z_1)
+
(x_2+y_2+z_2) =
0.
$$

Thus \(u+v \in W\).

Finally, let \(c \in \mathbb{R}\). Then

$$
c(x+y+z) = c \cdot 0 = 0,
$$

so

$$
cu \in W.
$$

Therefore \(W\) is a subspace of \(\mathbb{R}^3\).

Geometrically, this subspace is a plane through the origin.

## 18.6 Non-Examples

Many sets fail the subspace conditions in subtle ways.

### Example 1

Consider

$$
S =
\left\{
\begin{bmatrix}
x \\
y
\end{bmatrix}
:
xy = 0
\right\}.
$$

This set contains vectors lying on the coordinate axes.

Take

$$
u =
\begin{bmatrix}
1 \\
0
\end{bmatrix},
\qquad
v =
\begin{bmatrix}
0 \\
1
\end{bmatrix}.
$$

Both belong to \(S\), but

$$
u+v =
\begin{bmatrix}
1 \\
1
\end{bmatrix}
\notin S,
$$

since

$$
1 \cdot 1 \neq 0.
$$

Thus \(S\) is not closed under addition.

### Example 2

Consider

$$
T =
\left\{
\begin{bmatrix}
x \\
y
\end{bmatrix}
:
x \geq 0
\right\}.
$$

The vector

$$
\begin{bmatrix}
1 \\
0
\end{bmatrix}
\in T,
$$

but

$$
-1
\begin{bmatrix}
1 \\
0
\end{bmatrix} =
\begin{bmatrix}
-1 \\
0
\end{bmatrix}
\notin T.
$$

Thus \(T\) is not closed under scalar multiplication.

## 18.7 Trivial Subspaces

Every vector space \(V\) has at least two subspaces:

| Subspace | Description |
|---|---|
| \(\{0\}\) | Zero subspace |
| \(V\) | Whole space |

These are called the trivial subspaces.

The zero subspace contains only the zero vector. Its dimension is zero.

The whole space is always a subspace of itself.

## 18.8 Solution Spaces of Homogeneous Systems

One of the most important sources of subspaces comes from homogeneous linear equations.

Consider the system

$$
Ax = 0.
$$

The set of all solutions is called the null space or kernel of \(A\).

If \(u\) and \(v\) satisfy

$$
Au = 0
$$

and

$$
Av = 0,
$$

then

$$
A(u+v) =
Au + Av =
0 + 0 =
0.
$$

Thus the solution set is closed under addition.

Similarly,

$$
A(cv) =
cAv =
c0 =
0,
$$

so the solution set is closed under scalar multiplication.

Therefore the solution set is a subspace.

This fact is fundamental. Linear equations naturally produce vector spaces.

## 18.9 Column Space

Let \(A\) be an \(m \times n\) matrix with columns

$$
a_1, a_2, \ldots, a_n.
$$

The column space of \(A\) is the set of all linear combinations of the columns:

$$
\operatorname{Col}(A) =
\operatorname{span}(a_1,\ldots,a_n).
$$

Since spans are subspaces, the column space is a subspace of \(\mathbb{R}^m\).

The column space describes every vector that can be written as

$$
Ax.
$$

Thus the equation

$$
Ax = b
$$

has a solution exactly when

$$
b \in \operatorname{Col}(A).
$$

The column space measures the range of the linear transformation defined by \(A\).

## 18.10 Row Space

The row space of a matrix is the span of its row vectors.

If

$$
A =
\begin{bmatrix}
r_1 \\
r_2 \\
\vdots \\
r_m
\end{bmatrix},
$$

then

$$
\operatorname{Row}(A) =
\operatorname{span}(r_1,\ldots,r_m).
$$

The row space is a subspace of \(\mathbb{R}^n\).

Row space and column space are closely related. Later chapters show that they always have the same dimension.

## 18.11 Null Space

The null space of a matrix \(A\) is

$$
\operatorname{Null}(A) =
\{x : Ax = 0\}.
$$

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

The null space measures directions collapsed by the transformation \(x \mapsto Ax\).

If the null space contains only the zero vector, then the transformation is injective.

If the null space contains nonzero vectors, then information is lost under the transformation.

## 18.12 Span as a Subspace

Every span is a subspace.

Let

$$
W = \operatorname{span}(v_1,\ldots,v_k).
$$

By definition, every vector in \(W\) has the form

$$
c_1v_1 + \cdots + c_kv_k.
$$

If

$$
u = a_1v_1 + \cdots + a_kv_k
$$

and

$$
v = b_1v_1 + \cdots + b_kv_k,
$$

then

$$
u+v =
(a_1+b_1)v_1 + \cdots + (a_k+b_k)v_k.
$$

Thus \(u+v \in W\).

Similarly,

$$
cu =
(ca_1)v_1 + \cdots + (ca_k)v_k,
$$

so \(cu \in W\).

Therefore \(W\) is a subspace.

In fact, the span of a set is the smallest subspace containing that set.

## 18.13 Intersection of Subspaces

If \(U\) and \(W\) are subspaces of \(V\), then

$$
U \cap W
$$

is also a subspace of \(V\).

To prove this, let \(u,v \in U \cap W\). Then both vectors belong to both subspaces.

Since each subspace is closed under addition,

$$
u+v \in U
$$

and

$$
u+v \in W.
$$

Therefore

$$
u+v \in U \cap W.
$$

The same argument works for scalar multiplication.

Thus intersections preserve subspace structure.

## 18.14 Sum of Subspaces

Let \(U\) and \(W\) be subspaces of \(V\). Their sum is

$$
U + W =
\{u+w : u \in U,\ w \in W\}.
$$

The sum is also a subspace of \(V\).

This construction combines two vector spaces into a larger one.

For example, in \(\mathbb{R}^3\), if \(U\) is the \(xy\)-plane and \(W\) is the \(z\)-axis, then

$$
U + W = \mathbb{R}^3.
$$

## 18.15 Direct Sums

A sum

$$
V = U + W
$$

is called a direct sum if every vector in \(V\) can be written uniquely as

$$
v = u + w,
$$

where

$$
u \in U,
\qquad
w \in W.
$$

This happens exactly when

$$
U \cap W = \{0\}.
$$

Direct sums decompose vector spaces into independent pieces.

The notation is

$$
V = U \oplus W.
$$

This idea becomes important in eigenspaces, invariant subspaces, and canonical forms.

## 18.16 Subspaces Generated by Equations

Subspaces often arise as solution sets of linear equations.

For example,

$$
x - 2y + 3z = 0
$$

defines a subspace of \(\mathbb{R}^3\).

But

$$
x - 2y + 3z = 5
$$

does not.

The reason is that homogeneous equations preserve the origin, while nonhomogeneous equations shift the solution set away from the origin.

The first equation defines a plane through the origin. The second defines a translated plane.

## 18.17 Finite-Dimensional Subspaces

Every subspace of a finite-dimensional vector space is finite-dimensional.

If

$$
W \subseteq \mathbb{R}^n,
$$

then

$$
\dim(W) \leq n.
$$

A line through the origin in \(\mathbb{R}^3\) has dimension \(1\). A plane through the origin has dimension \(2\).

No subspace of \(\mathbb{R}^3\) can have dimension greater than \(3\).

This fact later leads to the dimension theorem.

## 18.18 Polynomial Subspaces

Consider the set

$$
W =
\{p(x) \in P_3 : p(1)=0\}.
$$

This is a subspace of \(P_3\).

If

$$
p(1)=0
$$

and

$$
q(1)=0,
$$

then

$$
(p+q)(1)=p(1)+q(1)=0.
$$

Also,

$$
(cp)(1)=cp(1)=0.
$$

Thus the subspace conditions hold.

Every polynomial satisfying \(p(1)=0\) has the factor

$$
x-1.
$$

Therefore

$$
W =
\operatorname{span}(x-1,\ x(x-1),\ x^2(x-1)).
$$

This subspace has dimension \(3\).

## 18.19 Function Subspaces

Function spaces contain many important subspaces.

The set of continuous functions on an interval forms a subspace of the space of all functions.

The set of differentiable functions forms another subspace.

The set of solutions of a linear differential equation also forms a subspace.

For example, the solutions of

$$
y'' + y = 0
$$

form the subspace

$$
\operatorname{span}(\cos x,\ \sin x).
$$

This connection between differential equations and vector spaces is fundamental in analysis and physics.

## 18.20 Summary

A subspace is a subset of a vector space that remains closed under vector addition and scalar multiplication.

The key ideas are:

| Concept | Meaning |
|---|---|
| Subspace | A vector space inside another vector space |
| Closure | Operations stay inside the set |
| Zero vector | Must belong to every subspace |
| Span | Smallest subspace containing given vectors |
| Null space | Solutions of \(Ax=0\) |
| Column space | Span of matrix columns |
| Row space | Span of matrix rows |
| Sum of subspaces | Combination of vector spaces |
| Direct sum | Unique decomposition into subspaces |

Subspaces organize vector spaces into smaller linear pieces. Much of linear algebra studies how these pieces interact, combine, and decompose under linear transformations.