Learning Outcomes
- Determine if a set of Euclidean vectors spans \(\IR^n\) by solving appropriate vector equations.
Section 2.2 Spanning Sets (EV2)
Subsection 2.2.1 Class Activities
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 2.2.2.
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 2.2.3.
How many vectors are required to span \(\IR^3\text{?}\)
- \(\displaystyle 1\)
- \(\displaystyle 2\)
- \(\displaystyle 3\)
- \(\displaystyle 4\)
- Infinitely Many
Fact 2.2.4.
At least \(n\) vectors are required to span \(\IR^n\text{.}\)
Activity 2.2.5.
Consider the question: Does every vector in \(\IR^3\) belong to \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{?}\)
(a)
Determine if \(\left[\begin{array}{c} 7 \\ -3 \\ -2 \end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{.}\)
(b)
Determine if \(\left[\begin{array}{c} 2 \\ 5 \\ 7 \end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{.}\)
(c)
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.
(d)
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 2.2.6.
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 2.2.7.
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 2.2.8.
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 \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)
Subsection 2.2.2 Videos
Exercises 2.2.3 Exercises
Exercises available at
https://teambasedinquirylearning.github.io/linear-algebra/2024e/exercises/#/bank/EV2/.Subsection 2.2.4 Mathematical Writing Explorations
Exploration 2.2.9.
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 2.2.10.
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 2.2.5 Sample Problem and Solution
Sample problem Example B.1.6.
