Vector Subspaces

Definition

A subspace of a Vector Space $\mathbb{V}$ is a Subset $\mathbb{W} \subseteq \mathbb{V}$ that is itself a vector space under the same operations.

For $\mathbb{W}$ to be a subspace, it must satisfy three properties:

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

Note: Properties 1 and 2 together mean closed under Linear Combinations!

Examples

In $\mathbb{R}^3$

1. Any line through the origin
2. Any plane through the origin
3. $\{\mathbf{0}\}$ (the zero subspace)
4. All of $\mathbb{R}^3$

Counter-Examples

1. A line not through the origin (fails property 3)
2. The unit circle (fails properties 1 and 2)
3. $\mathbb{R}^3$ without the origin (fails property 3)

Common Ways to Get Subspaces

1. Spans of vectors
2. Solution space to homogeneous systems
3. Intersection of subspaces
4. Kernel (null space) of a Linear Transformation

Why Do We Care?

Subspaces help us:

1. Break down complicated spaces into simpler pieces
2. Understand solution spaces to systems of equations
3. Study linear transformations through their kernels and images
4. Develop the notion of Dimension

Think of subspaces as "mini vector spaces" living inside bigger ones - they inherit all the nice properties of vector spaces but might be easier to work with!

Exercise

Prove that the Intersection of any collection of subspaces is itself a subspace. (Hint: Check the three properties directly)