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## Section2.4Subspaces (VS4)

### Subsection2.4.1Class Activities

#### Activity2.4.1.

Consider two non-colinear vectors in $$\IR^3\text{.}$$ If we look at all linear combinations of those two vectors (that is, their span), we end up with a plane within $$\IR^3\text{.}$$ Call this plane $$S\text{.}$$

##### (a)

Are all of the vectors in $$S$$ also in $$\IR^3\text{?}$$

##### (b)

Let $$\vec{z}$$ be the additive identity in $$\IR^3\text{.}$$ Is $$\vec{z} \in S\text{?}$$

##### (c)

For any unspecified $$\vec{u}, \vec{v} \in S\text{,}$$ is it the case that $$\vec{u} + \vec{v} \in S\text{?}$$

##### (d)

For any unspecified $$\vec{u} \in S$$ and $$c\in\IR\text{,}$$ is it the case that $$c\vec{u} \in S\text{?}$$

#### Definition2.4.2.

A subset of a vector space is called a subspace if it is a vector space on its own. The operations of addition and scalar from the parent vector space are inherited by the subspace.

#### Observation2.4.3.

Note the similarities between a planar subspace spanned by two non-colinear vectors in $$\IR^3\text{,}$$ and the Euclidean plane $$\IR^2\text{.}$$ While they are not the same thing (and shouldn't be referred to interchangably), algebraists call such similar spaces isomorphic; we'll learn what this means more carefully in a later chapter.

#### Activity2.4.5.

Let $$S=\setBuilder{\left[\begin{array}{c} x \\ y \\ z \end{array}\right]}{ x+2y+z=0}\text{.}$$

##### (a)

Let $$\vec{v}=\left[\begin{array}{c} x \\ y \\ z \end{array}\right]$$ and $$\vec{w} = \left[\begin{array}{c} a \\ b \\ c \end{array}\right]$$ be vectors in $$S\text{,}$$ so $$x+2y+z=0$$ and $$a+2b+c=0\text{.}$$ Show that $$\vec v+\vec w = \left[\begin{array}{c} x+a \\ y+b \\ z+c \end{array}\right]$$ also belongs to $$S$$ by verifying that $$(x+a)+2(y+b)+(z+c)=0\text{.}$$

##### (b)

Let $$\vec{v}=\left[\begin{array}{c} x \\ y \\ z \end{array}\right]\in S\text{,}$$ so $$x+2y+z=0\text{.}$$ Show that $$c\vec v=\left[\begin{array}{c}cx\\cy\\cz\end{array}\right]$$ also belongs to $$S$$ for any $$c\in\IR$$ by verifying an appropriate equation.

##### (c)

Is $$S$$ is a subspace of $$\IR^3\text{?}$$

#### Activity2.4.6.

Let $$S=\setBuilder{\left[\begin{array}{c} x \\ y \\ z \end{array}\right]}{ x+2y+z=4}\text{.}$$ Choose a vector $$\vec v=\left[\begin{array}{c} \unknown\\\unknown\\\unknown \end{array}\right]$$ in $$S$$ and a real number $$c=\unknown\text{,}$$ and show that $$c\vec v$$ isn't in $$S\text{.}$$ Is $$S$$ a subspace of $$\IR^3\text{?}$$

#### Remark2.4.7.

Since $$0$$ is a scalar and $$0\vec{v}=\vec{z}$$ for any vector $$\vec{v}\text{,}$$ a nonempty set that is closed under scalar multiplication must contain the zero vector $$\vec{z}$$ for that vector space.

Put another way, you can check any of the following to show that a nonempty subset $$W$$ isn't a subspace:

• Show that $$\vec 0\not\in W\text{.}$$

• Find $$\vec u,\vec v\in W$$ such that $$\vec u+\vec v\not\in W\text{.}$$

• Find $$c\in\IR,\vec v\in W$$ such that $$c\vec v\not\in W\text{.}$$

If you cannot do any of these, then $$W$$ can be proven to be a subspace by doing the following:

• Prove that $$\vec u+\vec v\in W$$ whenever $$\vec u,\vec v\in W\text{.}$$

• Prove that $$c\vec v\in W$$ whenever $$c\in\IR,\vec v\in W\text{.}$$

#### Activity2.4.8.

Consider these subsets of $$\IR^3\text{:}$$

\begin{equation*} R= \setBuilder{ \left[\begin{array}{c}x\\y\\z\end{array}\right]}{y=z+1} \hspace{2em} S= \setBuilder{ \left[\begin{array}{c}x\\y\\z\end{array}\right]}{y=|z|} \hspace{2em} T= \setBuilder{ \left[\begin{array}{c}x\\y\\z\end{array}\right]}{z=xy}\text{.} \end{equation*}
##### (a)

Show $$R$$ isn't a subspace by showing that $$\vec 0\not\in R\text{.}$$

##### (b)

Show $$S$$ isn't a subspace by finding two vectors $$\vec u,\vec v\in S$$ such that $$\vec u+\vec v\not\in S\text{.}$$

##### (c)

Show $$T$$ isn't a subspace by finding a vector $$\vec v\in T$$ such that $$2\vec v\not\in T\text{.}$$

#### Activity2.4.9.

Consider these subsets of $$M_{2 \times 2}\text{,}$$ the vector space of all $$2 \times 2$$ matrices with real entries. Show that each of these sets is or is not a subspace of $$M_{2 \times 2}\text{.}$$

##### (a)
\begin{equation*} \setBuilder{ \left[\begin{array}{cc}a&0\\0&b\end{array}\right]}{a,b \in \IR}\text{.} \end{equation*}
##### (b)
\begin{equation*} \setBuilder{ \left[\begin{array}{cc}a&0\\0&b\end{array}\right]}{a + b = 0}\text{.} \end{equation*}
##### (c)
\begin{equation*} \setBuilder{ \left[\begin{array}{cc}a&0\\0&b\end{array}\right]}{a + b = 5}\text{.} \end{equation*}
##### (d)
\begin{equation*} \setBuilder{ \left[\begin{array}{cc}a&c\\0&b\end{array}\right]}{a + b = 0, c \in \IR}\text{.} \end{equation*}

#### Activity2.4.10.

Let $$W$$ be a subspace of a vector space $$V\text{.}$$ How are $$\vspan W$$ and $$W$$ related?

1. $$\vspan W$$ may include vectors that aren't in $$W$$

2. $$W$$ may include vectors that aren't in $$\vspan W$$

3. $$W$$ and $$\vspan W$$ always contain the same vectors

### Subsection2.4.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/VS4.slides.html.

### Subsection2.4.5Mathematical Writing Explorations

#### Exploration2.4.12.

A square matrix $$M$$ is symmetric if, for each index $$i,j\text{,}$$ the entries $$m_{ij} = m_{ji}\text{.}$$ That is, the matrix is itself when reflected over the diagonal from upper left to lower right. Prove that the set of $$n \times n$$ symmetric matrices is a subspace of $$M_{n \times n}\text{.}$$

#### Exploration2.4.13.

The space of all real-valued function of one real variable is a vector space. First, define $$\oplus$$ and $$\odot$$ for this vector space. Check that you have closure (both kinds!) and show what the zero vector is under your chosen addition. Decide if each of the following is a subspace. If so, prove it. If not, provide the counterexample.
• The set of even functions, $$\{f:\mathbb{R} \rightarrow \mathbb{R}: f(-x) = f(x) \mbox{ for all } x\}\text{.}$$

• The set of odd functions, $$\{f:\mathbb{R} \rightarrow \mathbb{R}: f(-x) = -f(x) \mbox{ for all } x\}\text{.}$$

#### Exploration2.4.14.

Give an example of each of these, or explain why it's not possible that such a thing would exist.
• A nonempty subset of $$M_{2 \times 2}$$ that is not a subspace.

• A set of two vectors in $$\mathbb{R}^2$$ that is not a spanning set.

#### Exploration2.4.15.

Let $$V$$ be a vector space and $$S = \{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\}$$ a subset of $$V\text{.}$$ Show that the span of $$S$$ is a subspace. Is it possible that there is a subset of $$V$$ containing fewer vectors than $$S\text{,}$$ but whose span contains all of the vectors in the span of $$S\text{?}$$

### Subsection2.4.6Sample Problem and Solution

Sample problem Example B.1.8.

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