Learning Outcomes
- Determine if a set of Euclidean vectors spans \(\IR^n\) by solving appropriate vector equations.
Section 2.3 Spanning Sets (VS3)
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 \(\IR^n\text{.}\)
Activity 2.3.5.
Consider the question: Does every vector in \(\IR^3\) belong to \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}\text{?}\)
(a)
Determine if \(\left[\begin{array}{c} 2 \\ 5 \\ 7 \end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}\text{.}\)
(b)
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 \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}\text{.}\)
(c)
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.
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 2.3.9.
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 2.3.10.
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 2.3.11.
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
Subsection 2.3.3 Slideshow
Slideshow of activities available at
https://teambasedinquirylearning.github.io/linear-algebra/2023e/VS3.slides.html.Exercises 2.3.4 Exercises
Exercises available at
https://teambasedinquirylearning.github.io/linear-algebra/2023e/exercises/#/bank/VS3/.Subsection 2.3.5 Mathematical Writing Explorations
Exploration 2.3.12.
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 \(\mathbb{R}^3\)
Exploration 2.3.13.
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.
