Subspaces

Overview

A subspace is a subset of a vector space that is itself a vector space under the same operations. Subspaces are fundamental in understanding the structure of vector spaces.

Definition

A subset $W$ of a vector space $V$ is a subspace if:

1. The zero vector $\mathbf{0} \in W$
2. $W$ is closed under addition: if $\mathbf{u}, \mathbf{v} \in W$, then $\mathbf{u} + \mathbf{v} \in W$
3. $W$ is closed under scalar multiplication: if $\mathbf{u} \in W$ and $c \in F$, then $c\mathbf{u} \in W$

Subspace Test

To prove $W$ is a subspace of $V$, verify:

1. $\mathbf{0} \in W$ (non-empty and contains zero)
2. $\mathbf{u} + \mathbf{v} \in W$ for all $\mathbf{u}, \mathbf{v} \in W$ (closed under +)
3. $c\mathbf{u} \in W$ for all $\mathbf{u} \in W$, $c \in F$ (closed under scalar ×)

Alternatively, conditions 2 and 3 can be combined:

• For all $\mathbf{u}, \mathbf{v} \in W$ and scalars $c, d$: $c\mathbf{u} + d\mathbf{v} \in W$

Important Subspaces

Trivial Subspaces

Every vector space $V$ has two trivial subspaces:

• $\{\mathbf{0}\}$ - the zero subspace
• $V$ itself

Null Space (Kernel)

For an $m \times n$ matrix $A$, the null space is:

$\text{Null}(A) = \{\mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = \mathbf{0}\}$

Properties:

• Always a subspace of $\mathbb{R}^n$
• Contains all solutions to homogeneous system $A\mathbf{x} = \mathbf{0}$

Column Space (Image)

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

$\text{Col}(A) = \{A\mathbf{x} : \mathbf{x} \in \mathbb{R}^n\}$

Properties:

• Subspace of $\mathbb{R}^m$
• Spanned by columns of $A$
• Range of the linear transformation $\mathbf{x} \mapsto A\mathbf{x}$

Row Space

For matrix $A$:

$\text{Row}(A) = \text{Col}(A^T)$

Properties:

• Subspace of $\mathbb{R}^n$
• Spanned by rows of $A$

Examples

Example 1: Line Through Origin

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

Test:

1. $(0, 0) \in W$ since $0 = 2(0)$ ✓
2. If $(a, 2a), (b, 2b) \in W$, then $(a+b, 2a+2b) = (a+b, 2(a+b)) \in W$ ✓
3. If $(a, 2a) \in W$, then $c(a, 2a) = (ca, 2ca) \in W$ ✓

$W$ is a subspace of $\mathbb{R}^2$.

Example 2: Plane Through Origin

$W = \{(x, y, z) \in \mathbb{R}^3 : x + y + z = 0\}$

This is a subspace (verify using the test).

Example 3: Non-Subspace

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

Not a subspace because:

• $(0, 0) \notin W$ (since $0 \neq 0 + 1$)

Intersection of Subspaces

If $U$ and $W$ are subspaces of $V$, then $U \cap W$ is also a subspace of $V$.

Proof:

1. $\mathbf{0} \in U$ and $\mathbf{0} \in W$, so $\mathbf{0} \in U \cap W$
2. If $\mathbf{u}, \mathbf{v} \in U \cap W$, then $\mathbf{u} + \mathbf{v} \in U$ and $\mathbf{u} + \mathbf{v} \in W$, so $\mathbf{u} + \mathbf{v} \in U \cap W$
3. If $\mathbf{u} \in U \cap W$, then $c\mathbf{u} \in U$ and $c\mathbf{u} \in W$, so $c\mathbf{u} \in U \cap W$

Sum of Subspaces

The sum of subspaces $U$ and $W$:

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

This is also a subspace.