Subspace

A subspace is subset of a Vector Space that itself is a vector space. It must use the same addition and multiplication operations defined in it’s superset.

In other words, $W_{\mathbb{F}}$ is a subspace of the vector space $V_{\mathbb{F}}$ if and only if $W\underline{\subset }V$ and it uses the same multiplication and addition operators defined in $V_{\mathbb{F}}$. It must also be using the same Field, $\mathbb{F}$, that $V_{\mathbb{F}}$ is fixed over.

#ask Why is a vector space a subspace of itself?

Subspace Theorem §

Let $V_{\mathbb{F}}$ be a vector space fixed over the field $\mathbb{F}$ . A subset $W\underline{\subset }V$ is a subspace ($W_{\mathbb{F}}$) only if:

0. Contains the Zero Vector §

$W$ must contain the zero vector ${\vec{0}}_V$ from the set $V$

1. Closed under Vector Addition §

By definition of the binary operators, vector addition must be closed:

$$\forall \vec{u},\vec{v}\in W,\vec{u}+\vec{v}\in W$$

2. Closed under Scalar Multiplication §

Again, by definition of the binary operator multiplication:

$$\forall \vec{v}\in W,\alpha \in F,\alpha \times \vec{v}\in W$$

Examples §

For Real vector space ${\mathbb{R}}^2$, we have the subspaces:

• $\{\vec{0}\}$
• Any lines going through the origin (so it contains $\vec{0}$)
• ${\mathbb{R}}^2$ itself.

For ${\mathbb{R}}^3$, we have the subspaces:

• $\{\vec{0}\}$
• Any lines going through the origin (so it contains $\vec{0}$)
• Any planes intersecting the origin (so it contains $\vec{0}$)
• ${\mathbb{R}}^3$ itself.