TBIL-LA Spanning Sets (VS3)

Section 2.3 Spanning Sets (VS3)

Learning Outcomes

Determine if a set of Euclidean vectors spans \(\IR^n\) by solving appropriate vector equations.

Subsection 2.3.1 Class Activities

Observation 2.3.1.

Any single non-zero vector/number \(x\) in \(\IR^1\) spans \(\IR^1\text{,}\) since \(\IR^1=\setBuilder{cx}{c\in\IR}\text{.}\)

Activity 2.3.2.

How many vectors are required to span \(\IR^2\text{?}\) Sketch a drawing in the \(xy\) plane to support your answer.

\(\displaystyle 1\)
\(\displaystyle 2\)
\(\displaystyle 3\)
\(\displaystyle 4\)
Infinitely Many

Activity 2.3.3.

How many vectors are required to span \(\IR^3\text{?}\)

\(\displaystyle 1\)
\(\displaystyle 2\)
\(\displaystyle 3\)
\(\displaystyle 4\)
Infinitely Many

Fact 2.3.4.

At least \(n\) vectors are required to span \(\IR^n\text{.}\)

Figure 11. Failed attempts to span \(\IR^n\) by \(<n\) vectors

Activity 2.3.5.

Choose any vector \(\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]\) in \(\IR^3\) that is not in \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}\) by using technology to verify that \(\RREF \left[\begin{array}{cc|c}1&-2&\unknown\\-1&0&\unknown\\0&1&\unknown\end{array}\right] = \left[\begin{array}{cc|c}1&0&0\\0&1&0\\0&0&1\end{array}\right] \text{.}\) (Why does this work?)

Fact 2.3.6.

The set \(\{\vec v_1,\dots,\vec v_m\}\) fails to span all of \(\IR^n\) exactly when the vector equation

\begin{equation*} x_1 \vec{v}_1 + \cdots x_m\vec{v}_m = \vec{w} \end{equation*}

is inconsistent for some vector \(\vec{w}\text{.}\)

Note that this happens exactly when \(\RREF[\vec v_1\,\dots\,\vec v_m]\) has a non-pivot row of zeros.

\begin{equation*} \left[\begin{array}{cc}1&-2\\-1&0\\0&1\end{array}\right]\sim \left[\begin{array}{cc}1&0\\0&1\\0&0\end{array}\right] \end{equation*}

\begin{equation*} \Rightarrow \left[\begin{array}{cc|c}1&-2&a\\-1&0&b\\0&1&c\end{array}\right]\sim \left[\begin{array}{cc|c}1&0&0\\0&1&0\\0&0&1\end{array}\right] \text{for some choice of vector} \left[\begin{array}{c} a \\ b \\ c \end{array}\right] \text{.} \end{equation*}

Activity 2.3.7.

Consider the set of vectors \(S=\left\{ \left[\begin{array}{c}2\\3\\0\\-1\end{array}\right], \left[\begin{array}{c}1\\-4\\3\\0\end{array}\right], \left[\begin{array}{c}1\\7\\-3\\-1\end{array}\right], \left[\begin{array}{c}0\\3\\5\\7\end{array}\right], \left[\begin{array}{c}3\\13\\7\\16\end{array}\right] \right\}\) and the question “Does \(\IR^4=\vspan S\text{?}\)”

(a)

Rewrite this question in terms of the solutions to a vector equation.

(b)

Answer your new question, and use this to answer the original question.

Activity 2.3.8.

Consider the set of third-degree polynomials

\begin{align*} S=\{ &2x^3+3x^2-1, 2x^3+3, 3x^3+13x^2+7x+16,\\ &-x^3+10x^2+7x+14, 4x^3+3x^2+2 \} . \end{align*}

and the question “Does \(\P_3=\vspan S\text{?}\)”

(a)

Rewrite this question to be about the solutions to a polynomial equation.

(b)

Answer your new question, and use this to answer the original question.

Activity 2.3.9.

Consider the set of matrices

\begin{equation*} S = \left\{ \left[\begin{array}{cc} 1 & 3 \\ 0 & 1 \end{array}\right], \left[\begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array}\right], \left[\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array}\right] \right\} \end{equation*}

and the question “Does \(M_{2,2} = \vspan S\text{?}\)”

(a)

Rewrite this as a question about the solutions to a matrix equation.

(b)

Answer your new question, and use this to answer the original question.

Activity 2.3.10.

Let \(\vec{v}_1, \vec{v}_2, \vec{v}_3 \in \IR^7\) be three vectors, and suppose \(\vec{w}\) is another vector with \(\vec{w} \in \vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\}\text{.}\) What can you conclude about \(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{?}\)

\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \) is larger than \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)
\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} = \vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)
\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \) is smaller than \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)

Subsection 2.3.2 Videos

Figure 12. Video: Determining if a set spans a Euclidean space

Subsection 2.3.3 Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/VS3.slides.html.

Exercises 2.3.4 Exercises

Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/VS3/.

Subsection 2.3.5 Mathematical Writing Explorations

Exploration 2.3.11.

Construct each of the following, or show that it is impossible:

A set of 2 vectors that spans \(\mathbb{R}^3\)
A set of 3 vectors that spans \(\mathbb{R}^3\)
A set of 3 vectors that does not span \(\mathbb{R}^3\)
A set of 4 vectors that spans \(\mathbb{R}^3\)

For any of the sets you constructed that did span the required space, are any of the vectors a linear combination of the others in your set?

Exploration 2.3.12.

Based on these results, generalize this a conjecture about how a set of \(n-1, n\) and \(n+1\) vectors would or would not span \(\mathbb{R}^n\text{.}\)

Subsection 2.3.6 Sample Problem and Solution

Sample problem Example B.1.7.