Subspace

Subspace

A subspace is a subset of a vector space, while it also satisfies all the axioms of a vector space. Thus, a subspace is a vector space by itself.

Note: For subspaces of $\R^d$, it’s sufficient only to show the closure under addition and scalar multiplication, or closure under linear combinations.

Note: Every vector space, and therefore every subspace, must have a zero vector. In the case of $\R^d$, this is the origin.

Examples

Every geometric structure, such as lines, planes, circles, and so on, are sets of points. Some of these structures that contain the origin are subspaces.

Line through the origin

A line through the origin in $\R^d$ is a subspace of $\R^d$. A line in $\R^d$ can be represented by a reference point, $\bold{p}=\begin{bmatrix}x_{1} & x_{2}&\cdots&x_{d}\end{bmatrix}^T$, and a vector in the direction of the line, $\bold{v}=\begin{bmatrix}v_{1} & v_{2}&\cdots&v_{d}\end{bmatrix}^T$ . For example, the following is a subspace:

$$L=\{\bold{p+\alpha v}|\alpha \in \R\}$$ ...