Spans and subspaces

Vector spaces inside vector spaces

Examples of vector spaces and their subspaces

$\mathbb{R}^1$, $\mathbb{R}^2$, $\mathbb{R}^3$, $\mathbb{R}^4$, $\ldots, \mathbb{R}^n$ are examples of vector spaces (which we will define more carefully later).

It is possible to "embed" one vector space inside another. E.g., \[ \left\{ \begin{bmatrix} x \\ 0 \\ 0 \end{bmatrix} \;:\; x \in \mathbb{R} \right\} \] is a copy of $\mathbb{R}^1$ inside $\mathbb{R}^3$.

Similarly, we can embed $\mathbb{R}^2$ into $\mathbb{R}^3$ as the set \[ \left\{ \begin{bmatrix} x \\ y \\ 0 \end{bmatrix} \;:\; x,y \in \mathbb{R} \right\}. \]

These are examples of subspaces of $\mathbb{R}^3$.

Definition of subspaces of $\mathbb{R}^n$

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 $V$.

Equivalent description:

$V$ contains the zero vector $\mathbf{0}$.
$V$ is closed under vector addition, i.e., if $\mathbf{u},\mathbf{v} \in V$, then $\mathbf{u} + \mathbf{v} \in V$.
$V$ is closed under scalar multiplication, i.e., if $\mathbf{v} \in V$ and $r \in \mathbb{R}$, then $r \mathbf{v} \in V$.

Here, being "closed" under certain operations means starting from elements from the set and applying these operations, we will never leave the set.

An example

The set \[ V = \left\{ \begin{bmatrix} x \\ y \\ 0 \end{bmatrix} \;:\; x, y \in \mathbb{R} \right\} \] is the subset of vectors in $\mathbb{R}^3$ that has 0 as the third entry.

It is not hard to see $V$ satisfies the definition above, so $V$ is a subspace of $\mathbb{R}^3$. Indeed, it is exactly the $xy$-plane, which can be considered as a copy of $\mathbb{R}^2$ that is embedded inside $\mathbb{R}^3$.

Exercise

Determine which of the following sets form subspaces of $\mathbb{R}^2$: \[ \begin{aligned} S_1 &= \left\{ \begin{bmatrix} 0 \\ 0 \end{bmatrix} \right\} & S_3 &= \left\{ \begin{bmatrix} a \\ 2a \end{bmatrix} \;\mid\; a \in \mathbb{R} \right\} \\ S_2 &= \left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\} & S_4 &= \left\{ \begin{bmatrix} a \\ 7 \end{bmatrix} \;\mid\; a \in \mathbb{R} \right\} \\ S_5 &= \left\{ \; \right\} & S_6 &= \left\{ \begin{bmatrix} 3a \\ 2b \end{bmatrix} \;\mid\; a,b \in \mathbb{R} \right\} \end{aligned} \]

Review: span of vectors

Recall that given a set of vectors $\{ \mathbf{v}_1,\ldots,\mathbf{v}_m \} \subset \mathbb{R}^n$, its span is the set of all possible linear combinations of these vectors.

I.e., it is the set \[ \left\{ c_1 \mathbf{v}_1 + \cdots + c_m \mathbf{v}_m \;\mid\; c_1,\ldots,c_m \in \mathbb{R} \right\} \]

and we use the notation \[ \operatorname{span} \{ \mathbf{v}_1,\ldots,\mathbf{v}_m \}. \]

Spans as subspaces

Indeed, the span of any set $S \subset \mathbb{R}^n$, is a subspace of $\mathbb{R}^n$. (The converse is also true: every subspace in $\mathbb{R}^n$ is the span of some set of vectors in $\mathbb{R}^n$.)

In this case, we say the resulting subspace is the subspace spanned by $S$. It is easy to verify that $\operatorname{span} S$ is the smallest subspace that contains $S$.

Exercise. Explain why the these claims are true.

Exercises

What is the subspace spanned by \[ \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\} \;? \]

What about the subspace spanned by \[ \left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\} \;? \]

What about the subspace spanned by \[ \left\{ \begin{bmatrix} 0 \\ 0 \end{bmatrix} \right\} \;? \]

In $\mathbb{R}^3$, what is the subspace spanned by \[ \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 2 \\ 1 \end{bmatrix} \right\} \;? \]

Exercises

In the (virtual) RGB color space, what is the subspace spanned by \[ \left\{ \color{red}{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} } , \color{blue}{ \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} } \right\} \;? \]

Similarly, what about the subspace spanned by \[ \left\{ \begin{bmatrix} 0.5 \\ 0.5 \\ 0 \end{bmatrix} \begin{bmatrix} 0 \\ 0.5 \\ 0.5 \end{bmatrix} \right\} \;? \]