Section 2.3 Spanning Sets (V3)
Subsection 2.3.1 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 2.3.3.
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 2.3.5.
How many vectors are required to span \(\IR^3\text{?}\)
\(\displaystyle 1\)
\(\displaystyle 2\)
\(\displaystyle 3\)
\(\displaystyle 4\)
Infinitely Many
Fact 2.3.7.
At least \(n\) vectors are required to span \(\IR^n\text{.}\)
Activity 2.3.9.
Choose any vector \(\left[\begin{array}{c}\unknown\\\unknown\\\unknown\end{array}\right]\) in \(\IR^3\) that is not in \(\vspan\left\{\left[\begin{array}{c}1\\-1\\0\end{array}\right], \left[\begin{array}{c}-2\\0\\1\end{array}\right]\right\}\) by using technology to verify that \(\RREF \left[\begin{array}{cc|c}1&-2&\unknown\\-1&0&\unknown\\0&1&\unknown\end{array}\right] = \left[\begin{array}{cc|c}1&0&0\\0&1&0\\0&0&1\end{array}\right] \text{.}\) (Why does this work?)
Fact 2.3.10.
The set \(\{\vec v_1,\dots,\vec v_m\}\) fails to span all of \(\IR^n\) exactly when the vector 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.
Activity 2.3.11.
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 2.3.12.
Consider the set of third-degree polynomials
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 2.3.13.
Consider the set of matrices
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 2.3.14.
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
Exercises 2.3.3 Exercises
Exercises available at checkit.clontz.org 1 .
https://checkit.clontz.org/#/banks/tbil-la/V3/