A subspace is a subset of a vector space and acts as a vector space itself. If W is a subset (subspace) of V, the following properties must hold:
1.$0 ∈ W$
2.$u,v ∈ W$, then $u+v ∈ W$(closed under addition)
3.$u ∈ W$, then$\lambda u ∈ W$(closed under scalar multiplication)
Subspaces are defined by basis vectors, and the span (all linear combinations) of those form the subspace. So each element of the basis is linearly independent.
F.ex. in $\mathbb{R}^3$, we could f.ex. have these types of subspaces:
– the zero vector (a point)
– a line through the origin
– a plane through the origin (2d subspace)
– all of 3d space (needs three lin. indep. vectors as a basis)
Examples
| vector space $V$ | basis $B$ |
|---|---|
| $\mathbb{R}^m$ | ${e_1, e_2, \ldots, e_m }$ |
| $C(A)$ (subspace of $\mathbb{R}^m$) | independent columns of $A$ |
| $2 \times 2$ symmetric matrices (subspace of $\mathbb{R}^{2\times2}$) | $\left\{ \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} \right\}$ |
| $\mathbb{R}[x]$ (polynomials) | ${ x^i : i = 0, 1, \ldots }$ (infinite set) |
| ${0}$ (smallest vector space) | $\varnothing$ (empty set) |
Continue reading here: The fundamental subspaces, Chapter 4 notes