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Linear Algebra for Team-Based Inquiry Learning

2023 Edition

Steven Clontz	Drew Lewis
University of South Alabama

August 24, 2023

Section 2.2: Spanning Sets (EV2)

Observation 2.2.1

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

Activity 2.2.1 (~5 min)

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 2.2.2 (~5 min)

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

$\displaystyle 1$
$\displaystyle 2$
$\displaystyle 3$
$\displaystyle 4$
Infinitely Many

Fact 2.2.2

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

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

Activity 2.2.3 (~15 min)

Consider the question: Does every vector in $\IR^3$ belong to $\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{?}$

Part 1.

Determine if $\left[\begin{array}{c} 7 \\ -3 \\ -2 \end{array}\right]$ belongs to $\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{.}$

Activity 2.2.3 (~15 min)

Part 2.

Determine if $\left[\begin{array}{c} 2 \\ 5 \\ 7 \end{array}\right]$ belongs to $\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{.}$

Activity 2.2.3 (~15 min)

Part 3.

An arbitrary vector $\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]$ belongs to $\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}$ provided the equation

\begin{equation*} x_1\left[\begin{array}{c}1\\-1\\0\end{array}\right]+ x_2\left[\begin{array}{c}-2\\0\\1\end{array}\right]+ x_3\left[\begin{array}{c}-2\\-2\\2\end{array}\right]=\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right] \end{equation*}

has...

no solutions.
exactly one solution.
at least one solution.
infinitely-many solutions.

Activity 2.2.3 (~15 min)

Part 4.

We’re guaranteed at least one solution if the RREF of the corresponding augmented matrix has no contradictions; likewise, we have no solutions if the RREF corresponds to the contradiction $0=1\text{.}$ Given

\begin{equation*} \left[\begin{array}{ccc|c}1&-2&-2&\unknown\\-1&0&-2&\unknown\\0&1&2&\unknown\end{array}\right]\sim \left[\begin{array}{ccc|c}1&0&2&\unknown\\0&1&2&\unknown\\0&0&0&\unknown\end{array}\right] \end{equation*}

we may conclude that the set does not span all of $\IR^3$ because...

the row $[0\,1\,2\,|\,\unknown]$ prevents a contradiction.
the row $[0\,1\,2\,|\,\unknown]$ allows a contradiction.
the row $[0\,0\,0\,|\,\unknown]$ prevents a contradiction.
the row $[0\,0\,0\,|\,\unknown]$ allows a contradiction.

Fact 2.2.3

The set $\{\vec v_1,\dots,\vec v_m\}$ spans 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 consistent for every vector $\vec{w}\text{.}$

Likewise, 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 these two possibilities are decided based on whether or not $\RREF[\vec v_1\,\dots\,\vec v_m]$ has either all pivot rows, or at least one non-pivot row (a row of zeroes):

\begin{equation*} \left[\begin{array}{ccc|c}1&-2&-2\\-1&0&-2\\0&1&2\end{array}\right]\sim \left[\begin{array}{ccc|c}1&0&2\\0&1&2\\0&0&0\end{array}\right]\text{.} \end{equation*}

Activity 2.2.4 (~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 2.2.4 (~5 min)

Part 2.

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

Activity 2.2.5 (~5 min)

Let $\vec{v}_1, \vec{v}_2, \vec{v}_3 \in \IR^7$ be three Euclidean 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\} $ is the same as $\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}$
$\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} $ is smaller than $\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}$