Linear Algebra for Team-Based Inquiry Learning

2022 Edition

Steven Clontz Drew Lewis
University of South Alabama University of South Alabama

August 2, 2022

Section 2.3: Spanning Sets (VS3)

Observation 2.3.1

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

Figure 1. An \(\IR^1\) vector

Activity 2.3.2 (~5 min)

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

Figure 2. The \(xy\) plane \(\IR^2\)

  1. \(\displaystyle 1\)

  2. \(\displaystyle 2\)

  3. \(\displaystyle 3\)

  4. \(\displaystyle 4\)

  5. Infinitely Many

Activity 2.3.3 (~5 min)

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

Figure 3. \(\IR^3\) space

  1. \(\displaystyle 1\)

  2. \(\displaystyle 2\)

  3. \(\displaystyle 3\)

  4. \(\displaystyle 4\)

  5. Infinitely Many

Fact 2.3.4

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

Figure 4. Failed attempts to span \(\IR^n\) by \(<n\) vectors

Activity 2.3.5 (~15 min)

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 (~5 min)

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 \(\IR^4=\vspan S\text{?}\)”

Part 1.

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

Activity 2.3.7 (~5 min)

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 \(\IR^4=\vspan S\text{?}\)”

Part 2.

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

Activity 2.3.8 (~10 min)

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 \(\P_3=\vspan S\text{?}\)”

Part 1.

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

Activity 2.3.8 (~10 min)

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 \(\P_3=\vspan S\text{?}\)”

Part 2.

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

Activity 2.3.9 (~5 min)

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 \(M_{2,2} = \vspan S\text{?}\)”

Part 1.

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

Activity 2.3.9 (~5 min)

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 \(M_{2,2} = \vspan S\text{?}\)”

Part 2.

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

Activity 2.3.10 (~5 min)

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 \(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{?}\)

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