Learning Outcomes
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Determine if a set of Euclidean vectors spans \(\IR^m\) by solving appropriate vector equations.
Section 2.2 Spanning Sets (EV2)
Subsection 2.2.1 Warm Up
Activity 2.2.1.
Given a set of ingredients and a meal, a recipe is a list of amounts of each ingredient required to prepare the given meal.
(a)
Use the words vector and linear combination to create a new statement that is analogous to one above.
(b)
Building on your analogy, what role might the word span play?
Subsection 2.2.2 Class Activities
Observation 2.2.2.
Any single non-zero vector/number \(x\) in \(\IR^1\) spans \(\IR^1\text{,}\) since \(\IR^1=\setBuilder{cx}{c\in\IR}\text{.}\)
Activity 2.2.3.
How many vectors are required to span \(\IR^2\text{?}\) Sketch a drawing in the \(xy\) plane to support your answer.
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\(\displaystyle 1\)
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\(\displaystyle 2\)
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\(\displaystyle 3\)
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\(\displaystyle 4\)
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Infinitely Many
Activity 2.2.4.
How many vectors are required to span \(\IR^3\text{?}\)
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\(\displaystyle 1\)
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\(\displaystyle 2\)
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\(\displaystyle 3\)
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\(\displaystyle 4\)
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Infinitely Many
Fact 2.2.5.
Activity 2.2.6.
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)
Given this result, do we now know whether every vector in \(\mathbb R^3\) 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)
Determine if \(\left[\begin{array}{c} 0 \\ -4 \\ 3 \end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{.}\)
(d)
Determine if \(\left[\begin{array}{c} 2 \\ 5 \\ 7 \end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{.}\)
(e)
Fix the SageMath code below to visualize \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right],
\left[\begin{array}{c}-2\\0\\1\end{array}\right],\left[\begin{array}{c}-2\\-2\\2\end{array}\right]\right\}\text{.}\)
Activity 2.2.7.
We’d prefer a more methodical method to decide if every vector in \(\IR^n\) belongs to some spanning set, compared to the guess-and-check methods we used in Activity 2.2.6.
(a)
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...
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no solutions.
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exactly one solution.
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at least one solution.
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infinitely-many solutions.
(b)
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...
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the row \([0\,1\,2\,|\,\unknown]\) prevents a contradiction.
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the row \([0\,1\,2\,|\,\unknown]\) allows a contradiction.
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the row \([0\,0\,0\,|\,\unknown]\) prevents a contradiction.
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the row \([0\,0\,0\,|\,\unknown]\) allows a contradiction.
Fact 2.2.8.
The set \(\{\vec v_1,\dots,\vec v_n\}\) spans all of \(\IR^m\) exactly when the vector equation
\begin{equation*}
x_1 \vec{v}_1 + \cdots + x_n\vec{v}_n = \vec{w}
\end{equation*}
is consistent for every vector \(\vec{w}\in\IR^m\text{.}\)
Likewise, the set \(\{\vec v_1,\dots,\vec v_n\}\) fails to span all of \(\IR^m\) exactly when the vector equation
\begin{equation*}
x_1 \vec{v}_1 + \cdots + x_n\vec{v}_n = \vec{w}
\end{equation*}
is inconsistent for some vector \(\vec{w}\in\IR^m\text{.}\)
Note these two possibilities are decided based on whether or not the RREF of the vector equation’s coefficient matrix (that is, \(\RREF[\vec v_1\,\dots\,\vec v_n]\)) 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 2.2.9.
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 2.2.10.
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 \(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{?}\)
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\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \) is larger than \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)
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\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \) is the same as \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)
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\(\vspan \left\{ \vec{w}, \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \) is smaller than \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)
Subsection 2.2.3 Individual Practice
Activity 2.2.11.
One of our important results in this lesson is Fact 2.2.5, which states that a set of \(n\) vectors is required to span \(\IR^n\text{.}\) While we developed some geometric intuition for why this true, we did not prove it in class. Before coming to class next time, follow the steps outlined below to convince yourself of this fact using the concepts we learned in this lesson.
(a)
Let \(\{\vec{v}_1,\dots, \vec{v}_m\}\) be a set of vectors living in \(\IR^n\) and assume that \(m <n\text{.}\) How many rows and how many columns will the matrix \([\vec{v}_1\cdots \vec{v}_m]\) have?
(b)
Given no additional information about the vectors \(\vec{v}_1,\dots, \vec{v}_m\text{,}\) what is the maximum possible number of pivots in \(\RREF[\vec v_1\,\dots\,\vec v_m]\text{?}\)
(c)
Conclude that our given set of vector cannot span all of \(\IR^n\text{.}\)
Subsection 2.2.4 Videos
Subsection 2.2.5 Exercises
Exercises available at
https://library.tbil.org/2025/linear-algebra/exercises/#/bank/EV2/
Subsection 2.2.6 Mathematical Writing Explorations
Exploration 2.2.12.
Construct each of the following, or show that it is impossible:
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A set of 2 vectors that spans \(\mathbb{R}^3\)
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A set of 3 vectors that spans \(\mathbb{R}^3\)
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A set of 3 vectors that does not span \(\mathbb{R}^3\)
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A set of 4 vectors that spans \(\mathbb{R}^3\)
For any of the sets you constructed that did span the required vector space, are any of the vectors a linear combination of the others in your set?
Exploration 2.2.13.
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 2.2.7 Sample Problem and Solution
Sample problem Example B.1.6.
