Definition – Subspace. – Math-reference.net

Contents
1. Definition
2. Geometric meaning

Definition

Let $V$ be a vector space over a set $R$ . Given a set $S\subseteq{V}$ , we say that $S$ is a subspace of $V$ if, for every pair $a, b \in R$ and $\mathbf{x}, \mathbf{y} \in{S}$ , the element $\mathbf{z} = [a\times_{2}\mathbf{x}] +_2 [b\times_{2}\mathbf{y}]$ is also an element. That is, $\mathbf{z}\in{S}$ .

Geometric meaning

The definition simply states that, given a linear combination of two elements of $S$ , the resulting vector is also in $S$ . This definition relates to lines and planes in $\mathbb{R}^2$ and $\mathbb{R}^3$ . To see why this is true, we consider two cases. The first is that $\mathbf{x}$ and $\mathbf{y}$ are scalar multiples of each other, say $\mathbf{x} = c \times_2 \mathbf{y}$ . Then, we have

$\begin{aligned} a\times_{2}\mathbf{x} + b\times_{2}(a\times_{2}\mathbf{x}) &= (1+b)\times_{2}(a\times_{2}\mathbf{x}) \\ &=[(1+b)\times_{1}(a)]\times_{2}\mathbf{x} \end{aligned}$

Which is simply a constant times a vector, which is the equation of a straight line in $\mathbb{R}^2$ and $\mathbb{R}^3$ . Now, if the two vectors are not scalar multiples of each other, then we have the equation of a plane, which applies in both $\mathbb{R}^2$ and $\mathbb{R}^3$ . To see this, imagine we were to fix $a$ in $[a\times_{2}\mathbf{x}] +_2 [b\times_{2}\mathbf{y}]$ , then we have the equation of a line that intersects the point $a\times_{2}\mathbf{x}$ for all $a\in{\mathbb{R}}$ . I.e, we end up with a line for which each point on said line is a point on another line, with the equation of the line intersecting the first line at each point having the same gradient vector.

Background

In fact, we needn’t specify constants in the criteria for a subspace, as this is actually a consequence (via induction) of a simpler condition, that

$\forall \mathbf{x},\mathbf{y} \in S, ~\mathbf{x} + \mathbf{y} \in S$ .

Furthermore, a subspace is actually a subset $S\subseteq{V}$ such that $S$ is itself a vector space. But most of the conditions are fulfilled by the fact that $S$ is a subset of $V$ , and the remaining criteria that are fulfilled are covered by $\mathbf{x} + \mathbf{y} \in S$ .