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Linear Algebra for Team-Based Inquiry Learning

2023 Early Edition

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

December 22, 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 $\IR^1=\setBuilder{cx}{c\in\IR}\text{.}$

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.

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

Activity 2.3.3 (~5 min)

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

Activity 2.3.5 (~15 min)

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

Part 1.

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

Activity 2.3.5 (~15 min)

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

Part 2.

Write a system of equations to determine if the arbitrary vector $\left[\begin{array}{c} y_1 \\ y_2 \\ y_3 \end{array}\right]$ belongs to $\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}\text{.}$

Activity 2.3.5 (~15 min)

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

Part 3.

Write down the simpler system of equations that results from computing the RREF of the corresponding augmented matrix.

Observation 2.3.6

The vector 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] = \left[\begin{array}{c} y_1 \\ y_2 \\ y_3 \end{array}\right] \end{equation*}

simplifies to the system of equations

\begin{alignat*}{6} x_1 & & &+ y_2 & &= 0\\ & x_2 & & &-y_3 &= 0\\ & & y_1 &+y_2 & +2y_3 & = 0\text{.} \end{alignat*}

This last equation gives a relation among the components of the vector $\left[\begin{array}{c} y_1 \\ y_2 \\ y_3 \end{array}\right]$ that must be satisfied for it to belong to the span. Thus, we can conclude that $\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}$ is not all of $\IR^3\text{,}$ as we can find some vector that does not satisfy the equation $y_1+y_2+2y_3=0\text{.}$

Fact 2.3.7

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*}

as then $\RREF\left[\begin{array}{ccccc}1&-2& -1 & 0 & 0\\-1&0& 0 & -1 & 0 \\0&1& 0 & 0 & -1\end{array}\right] $ must have a pivot in one of the last three columns.

Conversely, if $\RREF[\vec v_1\,\dots\,\vec v_m]$ has a pivot in each row, there will not be a relation solely among $a, b, \text{and}\ c\text{,}$ in which case $\vspan\left\{\vec{v}_1,\dots, \vec{v}_m\right\} = \IR^n\text{.}$

Activity 2.3.8 (~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 2.3.8 (~5 min)

Part 2.

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

Activity 2.3.9 (~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 2.3.9 (~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 2.3.10 (~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 2.3.10 (~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 2.3.11 (~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 $\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}$