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Section 2.4 Subspaces (V4)

Definition 2.4.1.

A subset of a vector space is called a subspace if it is a vector space on its own.

For example, the span of these two vectors forms a planar subspace inside of the larger vector space \(\IR^3\text{.}\)

Figure 2.4.2. A subspace of \(\IR^3\)

Activity 2.4.4.

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 2.4.5.

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 2.4.6.

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 2.4.7.

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 2.4.8.

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

  1. \(\vspan W\) is bigger than \(W\)

  2. \(\vspan W\) is the same as \(W\)

  3. \(\vspan W\) is smaller than \(W\)

Subsection 2.4.1 Videos

Figure 2.4.10. Video: Determining if a subset of a vector space is a subspace

Exercises 2.4.2 Exercises

Exercises available at checkit.clontz.org 1 .

https://checkit.clontz.org/#/banks/tbil-la/V4/