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7.4. Subspaces and dimension

Eigenspaces are examples of subspaces, because the linear combination of any two vectors in the space is again in the space. An equivalent definition is given below. For us, the only vector space we will consider is ${\mathbb{R}}^{n}$ , for some integer $n\geq 1$ , which consists of column vectors (see Definition 1.1.4). The axiomatic definition of an abstract vector space is given in MATH220, and is not needed here.

Definition 7.4.1.

A subset of vectors, $W\subset{\mathbb{R}}^{n}$ , is called a subspace, if the following three axioms are satisfied:

S1

$0\in W$ ,
S2

$w_{1}+w_{2}\in W$ , for any $w_{1},w_{2}\in W$ ,
S3

$\lambda w\in W$ , for any $\lambda\in{\mathbb{R}}$ and $w\in W$ .

Example 7.4.2.

1. (a)
  
  The following subsets are subspaces, since all three axioms are satisfied:
  
  $\left\{\begin{pmatrix}2x\\ 0\end{pmatrix}\,|\,x\in{\mathbb{R}}\right\}\subset{\mathbb{R}}^{2},\quad\quad% \left\{\begin{pmatrix}2x+y\\ y\\ -x+3y\end{pmatrix}\,|\,x,y\in{\mathbb{R}}\right\}\subset{\mathbb{R}}^{3}.$
2. (b)
  
  The following subsets are not subspaces:
  
  $\left\{\begin{pmatrix}2x\\ 1\end{pmatrix}\,|\,x\in{\mathbb{R}}\right\}\subset{\mathbb{R}}^{2},\quad\quad% \left\{\begin{pmatrix}2x+y\\ y\\ -x^{2}+3y\end{pmatrix}\,|\,x,y\in{\mathbb{R}}\right\}\subset{\mathbb{R}}^{3}.$
  
  The subset on the left does not contain the zero vector, so S1 is not satisfied. For the subset on the right, notice that $w_{1}:=\begin{pmatrix}2\\ 0\\ -1\end{pmatrix}\in W$ , but $2w_{1}$ is not in $W$ , so S3 is not satisfied.
3. (c)
  
  If $A\in\operatorname{M}_{{n}}({{\mathbb{R}}})$ is a square matrix, and $\lambda\in{\mathbb{R}}$ is an eigenvalue of that matrix, then the eigenspace $V_{\lambda}$ is a subspace of ${\mathbb{R}}^{n}$ .
4. (d)
  
  The subset consisting of only the zero vector is a subspace (called the zero subspace).

In ${\mathbb{R}}^{2}$ , there are only three types of subspaces: the zero subspace, straight lines through the origin, and all of ${\mathbb{R}}^{2}$ . Similarly, in ${\mathbb{R}}^{3}$ , there are four types of subspaces: the zero subspace, straight lines through the origin, flat planes through the origin, and all of ${\mathbb{R}}^{3}$ .

A standard way of constructing a subspace is as follows.

Definition 7.4.3.

Given a set of vectors $\{v_{1},\cdots,v_{r}\}$ in ${\mathbb{R}}^{n}$ , where $r\geq 1$ is the number of vectors, we define the following subset of vectors:

\operatorname{span}\{{v_{1},\cdots,v_{r}}\}:=\{a_{1}v_{1}+\cdots+a_{r}v_{r}\;|% \;a_{i}\in{\mathbb{R}}\}\subset{\mathbb{R}}^{n}.

This is called the span of $\{v_{1},\cdots,v_{r}\}$ . It is the collection of all linear combinations of the vectors $v_{i}$ . It is always a subspace of ${\mathbb{R}}^{n}$ , and in fact all subspace can be written in this form. In this case, we say $v_{1},\cdots,v_{r}$ span the subspace.

Example 7.4.4.

The two subspaces from Example 7.4(i) can be expressed as:

$\operatorname{span}\{{\begin{pmatrix}2\\ 0\end{pmatrix}}\}\subset{\mathbb{R}}^{2},\quad\quad\operatorname{span}\{{% \begin{pmatrix}2\\ 0\\ -1\end{pmatrix},\begin{pmatrix}1\\ 1\\ 3\end{pmatrix}}\}\subset{\mathbb{R}}^{3}.$

Example 7.4.5.

1. (a)
  
  Consider the two eigenspaces from Example 7.2. These are subspaces, (since all eigenspaces are subspaces) and they may be rewritten as follows:
  
  $V_{4}=\operatorname{span}\{{\begin{pmatrix}5\\ 1\end{pmatrix}}\},\quad V_{-2}=\operatorname{span}\{{\begin{pmatrix}1\\ -1\end{pmatrix}}\}.$
2. (b)
  
  Consider the three eigenspaces found in Example 7.2. These subspaces may be rewritten as follows:
  
  $V_{0}=\operatorname{span}\{{\begin{pmatrix}0\\ 1\\ 1\end{pmatrix}}\},\quad V_{2}=\operatorname{span}\{{\begin{pmatrix}1\\ 2\\ 2\end{pmatrix}}\},\quad V_{-1}=\operatorname{span}\{{\begin{pmatrix}1\\ -3\\ -2\end{pmatrix}}\}.$

Notice that there are many ways of choosing vectors which span a subspace. For example, $\operatorname{span}\{{\begin{pmatrix}2\\ 0\end{pmatrix}}\}=\operatorname{span}\{{\begin{pmatrix}1\\ 0\end{pmatrix}}\}=\operatorname{span}\{{\begin{pmatrix}5\\ 0\end{pmatrix}}\}$ . Only one vector is needed to span this subspace, but that vector is not unique.

Definition 7.4.6.

The dimension of a subspace $W$ is the minimum number of vectors which span the subspace. This number is denoted $\dim W$ . If $\{v_{1},\cdots,v_{r}\}$ is a minimal set of spanning vectors (so $r=\dim W$ ), then we call this set a basis (plural: bases).

There is a general procedure to determine the dimension of a subspace, but this will be left for MATH220. In ${\mathbb{R}}^{3}$ it is much easier, since subspaces can only have dimension $0,1,2,$ or $3$ . A subspace has dimension 0 when it consists of just the zero vector, and dimension 3 when it is all of ${\mathbb{R}}^{3}$ . It has dimension 1 if it can be spanned by a single non-zero vector, like most of the examples above. So to prove that a subspace of ${\mathbb{R}}^{3}$ is 2 dimensional, one just needs to show that it isn’t 0, 1, or 3 dimensional. For an example, see Exercise 7.5.3(vi).

Example 7.4.7.

Let $W$ be the subspace spanned by the vectors

$u=\begin{pmatrix}1\\ 1\end{pmatrix},\quad v=\begin{pmatrix}1\\ -1\end{pmatrix}\quad\mbox{and}\quad w=\begin{pmatrix}0\\ 1\end{pmatrix}.$

Then $\operatorname{span}\{{u,v,w}\}=\mathbb{R}^{2}$ so $\dim W=2$ . To see this we note that any vector $(x,y)\in\mathbb{R}^{2}$ can be written as a linear combination of the vectors $u$ and $v$ :

$\begin{pmatrix}x\\ y\end{pmatrix}=\frac{x+y}{2}\begin{pmatrix}1\\ 1\end{pmatrix}+\frac{x-y}{2}\begin{pmatrix}1\\ -1\end{pmatrix}.$

That is, $\{u,v\}$ forms a basis for $W$ .