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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)

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)

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

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

$\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\}$ 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 $\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}$

Linear Algebra for Team-Based Inquiry Learning 2022 Edition Steven Clontz Drew Lewis University of South Alabama University of South Alabama August 2, 2022

Linear Algebra for Team-Based Inquiry Learning

2022 Edition

August 2, 2022

Section 2.3: Spanning Sets (VS3)

Observation 2.3.1

Activity 2.3.2 (~5 min)

Activity 2.3.3 (~5 min)

Fact 2.3.4

Activity 2.3.5 (~15 min)

Fact 2.3.6

Activity 2.3.7 (~5 min)

Activity 2.3.7 (~5 min)

Activity 2.3.8 (~10 min)

Activity 2.3.8 (~10 min)

Activity 2.3.9 (~5 min)

Activity 2.3.9 (~5 min)

Activity 2.3.10 (~5 min)