Section 2.3 Spanning Sets (V3)
Observation 2.3.1.
Any single non-zero vector/number \(x\) in \(\IR^1\) spans \(\IR^1\text{,}\) since \(\IR^1=\setBuilder{cx}{c\in\IR}\text{.}\)
\begin{tikzpicture}
\draw[<->] (-3,0) -- (3,0);
\draw[thick,->,blue] (0,0) -- (2,0) node[above] {x};
\draw (0,-0.2) -- (0,0.2) node[above] {0};
\end{tikzpicture}
Activity 2.3.1.
How many vectors are required to span \(\IR^2\text{?}\) Sketch a drawing in the \(xy\) plane to support your answer.
\begin{tikzpicture}[scale=0.5]
\draw[<->] (-4,0) -- (4,0);
\draw[<->] (0,-4) -- (0,4);
\end{tikzpicture}
\(\displaystyle 1\)
\(\displaystyle 2\)
\(\displaystyle 3\)
\(\displaystyle 4\)
Infinitely Many
Activity 2.3.2.
How many vectors are required to span \(\IR^3\text{?}\)
\begin{tikzpicture}[x={(210:0.8cm)}, y={(0:1cm)}, z={(90:1cm)},scale=0.4]
\draw[->] (0,0,0) -- (6,0,0);
\draw[->] (0,0,0) -- (0,6,0);
\draw[->] (0,0,0) -- (0,0,6);
\end{tikzpicture}
\(\displaystyle 1\)
\(\displaystyle 2\)
\(\displaystyle 3\)
\(\displaystyle 4\)
Infinitely Many
Fact 2.3.2.
At least \(n\) vectors are required to span \(\IR^n\text{.}\)
\begin{tikzpicture}[scale=0.5]
\draw[<->] (-4,0) -- (4,0);
\draw[<->] (0,-4) -- (0,4);
\draw[blue!50] (2,-4) -- (-2,4);
\draw[thick,blue,->] (0,0) -- (1,-2);
\end{tikzpicture}
\begin{tikzpicture}[x={(210:0.8cm)}, y={(0:1cm)}, z={(90:1cm)},scale=0.4]
\draw[->] (0,0,0) -- (6,0,0);
\draw[->] (0,0,0) -- (0,6,0);
\draw[->] (0,0,0) -- (0,0,6);
\draw[fill=purple!20,fill opacity=0.5]
(-2,-2,2) -- (6,-2,-2) -- (2,2,-2) -- (-6,2,2) -- (-2,-2,2);
\draw[thick,blue,->] (0,0,0) -- (1,-1,0);
\draw[thick,red,->] (0,0,0) -- (-2,0,1);
\end{tikzpicture}
Activity 2.3.3.
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 2.3.3.
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 2.3.4.
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 2.3.5.
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 2.3.6.
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 2.3.7.
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 \(\vspan \left\{ \vec{v}_1, \vec{v}_2, \vec{v}_3 \right\} \text{.}\)
Exercises 2.3.1 Exercises
1.
Consider the statement
Claim 2.3.4.
\(\left\{ \left[\begin{array}{c}
3 \\
-5 \\
-5 \\
0
\end{array}\right] , \left[\begin{array}{c}
3 \\
-3 \\
3 \\
4
\end{array}\right] , \left[\begin{array}{c}
-4 \\
-3 \\
2 \\
4
\end{array}\right] , \left[\begin{array}{c}
-10 \\
7 \\
12 \\
4
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)
- Write an equivalent statement using a vector equation.
- Explain why your statement is true or false.
- The statement
Claim 2.3.5.
\(\left\{ \left[\begin{array}{c} 3 \\ -5 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -10 \\ 7 \\ 12 \\ 4 \end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)Claim 2.3.6.
\(x_{1} \left[\begin{array}{c} 3 \\ -5 \\ -5 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ -3 \\ 2 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -10 \\ 7 \\ 12 \\ 4 \end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\) - The set of vectors \(\left\{ \left[\begin{array}{c} 3 \\ -5 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -10 \\ 7 \\ 12 \\ 4 \end{array}\right] \right\} \) does not span \(\mathbb{R}^4\text{.}\)
2.
Consider the statement
Claim 2.3.7.
\(\left\{ \left[\begin{array}{c}
4 \\
-3 \\
1 \\
2
\end{array}\right] , \left[\begin{array}{c}
-5 \\
1 \\
4 \\
-4
\end{array}\right] , \left[\begin{array}{c}
-3 \\
4 \\
-4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-3 \\
2 \\
-3
\end{array}\right] , \left[\begin{array}{c}
1 \\
-1 \\
3 \\
0
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)
- Write an equivalent statement using a vector equation.
- Explain why your statement is true or false.
- The statement
Claim 2.3.8.
\(\left\{ \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ 0 \end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)Claim 2.3.9.
\(x_{1} \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 1 \\ 4 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ 4 \\ -4 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -3 \end{array}\right] + x_{5} \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ 0 \end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\) - The set of vectors \(\left\{ \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ 0 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\text{.}\)
3.
Consider the statement
Claim 2.3.10.
\(\left\{ \left[\begin{array}{c}
2 \\
-4 \\
-3 \\
1
\end{array}\right] , \left[\begin{array}{c}
4 \\
-1 \\
-3 \\
-1
\end{array}\right] , \left[\begin{array}{c}
0 \\
-2 \\
1 \\
2
\end{array}\right] , \left[\begin{array}{c}
-5 \\
3 \\
-5 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-3 \\
-5 \\
-2 \\
-4
\end{array}\right] , \left[\begin{array}{c}
3 \\
-1 \\
-5 \\
0
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)
- Write an equivalent statement using a vector equation.
- Explain why your statement is true or false.
- The statement
Claim 2.3.11.
\(\left\{ \left[\begin{array}{c} 2 \\ -4 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ -5 \\ 0 \end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)Claim 2.3.12.
\(x_{1} \left[\begin{array}{c} 2 \\ -4 \\ -3 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -1 \\ -3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -2 \\ 1 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ 3 \\ -5 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ -5 \\ -2 \\ -4 \end{array}\right] + x_{6} \left[\begin{array}{c} 3 \\ -1 \\ -5 \\ 0 \end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\) - The set of vectors \(\left\{ \left[\begin{array}{c} 2 \\ -4 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -5 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ -5 \\ 0 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\text{.}\)
4.
Consider the statement
Claim 2.3.13.
\(\left\{ \left[\begin{array}{c}
2 \\
2 \\
0 \\
-4
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-4 \\
4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
-3 \\
-4 \\
-4 \\
-1
\end{array}\right] , \left[\begin{array}{c}
1 \\
0 \\
8 \\
0
\end{array}\right] , \left[\begin{array}{c}
-5 \\
4 \\
0 \\
-3
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)
- Write an equivalent statement using a vector equation.
- Explain why your statement is true or false.
- The statement
Claim 2.3.14.
\(\left\{ \left[\begin{array}{c} 2 \\ 2 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 8 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -3 \end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)Claim 2.3.15.
\(x_{1} \left[\begin{array}{c} 2 \\ 2 \\ 0 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -4 \\ 4 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -4 \\ -4 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ 0 \\ 8 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -3 \end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\) - The set of vectors \(\left\{ \left[\begin{array}{c} 2 \\ 2 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 8 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -3 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\text{.}\)
5.
Consider the statement
Claim 2.3.16.
\(\left\{ \left[\begin{array}{c}
-1 \\
-1 \\
-3 \\
3
\end{array}\right] , \left[\begin{array}{c}
4 \\
-3 \\
-5 \\
0
\end{array}\right] , \left[\begin{array}{c}
3 \\
-2 \\
-2 \\
-5
\end{array}\right] , \left[\begin{array}{c}
-2 \\
-5 \\
1 \\
3
\end{array}\right] , \left[\begin{array}{c}
1 \\
4 \\
-2 \\
-3
\end{array}\right] , \left[\begin{array}{c}
-2 \\
1 \\
1 \\
2
\end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)
- Write an equivalent statement using a vector equation.
- Explain why your statement is true or false.
- The statement
Claim 2.3.17.
\(\left\{ \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 2 \end{array}\right] \right\} \)\(\mathbb{R}^4\text{.}\)Claim 2.3.18.
\(x_{1} \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -3 \\ -5 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ -2 \\ -2 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ -5 \\ 1 \\ 3 \end{array}\right] + x_{5} \left[\begin{array}{c} 1 \\ 4 \\ -2 \\ -3 \end{array}\right] + x_{6} \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 2 \end{array}\right] =\vec{v}\)\(\vec{v}\)\(\mathbb{R}^4\text{.}\) - The set of vectors \(\left\{ \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 2 \end{array}\right] \right\} \) spans \(\mathbb{R}^4\text{.}\)
Additional exercises available at checkit.clontz.org.