Math1231 Subspaces

Definition

If S and V are vector spaces and S is a subset of V, we say that S is a subspace of V.

Necessary Conditions

  • S contains the zero vector
    • This also checks that it's non-empty
  • $\text {If } v1, v2 ∈ S \text { then } v1 + v2 ∈ S \text { and } λv1 ∈ S,\text { for all scalars } λ$
    • i.e. closed under vector/scalar addition/multiplication

Or:
If it's non empty then it's a subspace if: $λu1+µu2 ∈ S \text { for all } u1, u2 ∈ S \text { and } λ, µ \text { scalars}$

Example

(1)
\begin{align} \begin {array} {l} \text {Say } \mathbb{R}^2 \text { is a vector space} \\ \\ \text {Let } S = {(x,y): y>x} \\ \text {Let } T = {(x,y): y=0} \\ \\ S \text { is part of } \mathbb{R}^2, \text { but is not a vector space, as not all vector axioms hold} \\ \text {e.g. axiom 6, closure under scalar multiplication, doesn't hold with a negative number:} \\ \\ (1,2) \rightarrow (-1,-2), y > x \rightarrow x > y \\ \\ T \text { is a vector space in } \mathbb{R}^2, \text { as all axioms hold, and a subspace of } \mathbb{R}^2 \end {array} \end{align}