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Section 2.4 Subspaces (VS4)

Subsection 2.4.1 Class Activities

Activity 2.4.1.

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 \(c\vec{u} \in S\text{?}\)

Definition 2.4.2.

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.

Observation 2.4.3.

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.

A planar subset of \(\IR^3\) compared with the plane \(\IR^2\text{.}\)
Figure 13. A planar subset of \(\IR^3\) compared with the plane \(\IR^2\text{.}\)

Activity 2.4.5.

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 \(\IR^3\text{?}\)

Activity 2.4.6.

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 \(\IR^3\text{?}\)

Remark 2.4.7.

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 \(c\in\IR,\vec v\in W\text{.}\)

Activity 2.4.8.

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 \(2\vec v\not\in T\text{.}\)

Activity 2.4.9.

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*}

Activity 2.4.10.

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

Subsection 2.4.2 Videos

Figure 14. Video: Showing that a subset of a vector space is a subspace
Figure 15. Video: Showing that a subset of a vector space is not a subspace

Subsection 2.4.3 Slideshow

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

Exercises 2.4.4 Exercises

Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/VS4/.

Subsection 2.4.5 Mathematical Writing Explorations

Exploration 2.4.12.

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 \(M_{n \times n}\text{.}\)

Exploration 2.4.13.

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, \(\{f:\mathbb{R} \rightarrow \mathbb{R}: f(-x) = -f(x) \mbox{ for all } x\}\text{.}\)

Exploration 2.4.14.

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.

Exploration 2.4.15.

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 \(S\text{?}\)

Subsection 2.4.6 Sample Problem and Solution

Sample problem Example B.1.8.

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