We have used the word “space” in our previous discussion. In particular, we mentioned the concept of “row space” and “column space” associated with a matrix. These are examples of “subspaces” of $\mathbb{R}^n$, which are themselves examples of vector spaces (a more general subject that we will explore soon).
A subset $V$ of $\mathbb{R}^n$ is called a subspace of $\mathbb{R}^n$ if $V$ is nonempty and $V$ is closed under linear combination, i.e., for any $\mathbf{u},\mathbf{v} \in V$ and $a,b \in \mathbb{R}$, the linear combination $a \mathbf{u} + b \mathbf{v}$ remains in $V$.
Another equivalent definition for a subset $V$ being a subspace is that
Here, being “closed” under certain operations means starting from elements from the set and applying these operations, we will never leave the set.
Exercise. Determine if the following sets form subspaces of $\mathbb{R}^2$, and explain your reasoning.
\[\begin{aligned} S_1 &= \left\{ \begin{bmatrix} 0 \\ 0 \end{bmatrix} \right\} & S_3 &= \left\{ \begin{bmatrix} a \\ 2a \end{bmatrix} \;\mid\; a \in \mathbb{R} \right\} \\ S_2 &= \left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\} & S_4 &= \left\{ \begin{bmatrix} a \\ 7 \end{bmatrix} \;\mid\; a \in \mathbb{R} \right\} \\ S_5 &= \left\{ \; \right\} & S_6 &= \left\{ \begin{bmatrix} 3a \\ 2b \end{bmatrix} \;\mid\; a,b \in \mathbb{R} \right\} \end{aligned}\]In this course, most of the subspaces of $\mathbb{R}^n$ we will see are created as spans of sets of vectors.
One important fact is that given any set $S \subset \mathbb{R}^n$, the span of $S$ is a subspace of $\mathbb{R}^n$.
In this case, we say the resulting subspace space is the vector space spanned by $S$. It is easy to verify that $\operatorname{span} S$ is the smallest vector space that contains $S$.
So far, we have been careful using the term “subspace of $\mathbb{R}^n$” instead of the more familiar term “vector space”. All the subspaces we mentioned are indeed vector spaces, but vector spaces are much more general concepts. Simply put, all subspaces of $\mathbb{R}^n$ are indeed vector spaces, but not all vector spaces are subspaces of $\mathbb{R}^n$.
In the next sections, we will see more abstract vector spaces whose elements cannot be written as row or column vectors.