Question 3.19.
Find a way to write \(\vec{b}=\colvec{2\\ 4}\) as a linear combination of \(\vec{v_1}=\colvec{1\\ 1}\) and \(\vec{v_2}=\colvec{-1\\ 1}\) or explain why it is not possible to do so?
Section 3.3 Span
Recall that a linear combination of the set \(\{\vec{v_1},..\vec{v_k} \}\) is a vector of the form
\begin{equation*}
\sum_{i=1}^k c_i \vec{v_i} = c_1 \vec{v_1} + c_2 \vec{v_2}+...+c_k \vec{v_k}\text{.}
\end{equation*}
Note that some of the \(c_i\) may be zero. In other words, not every vector in a set needs to be part of a linear combination from that set.
Exercise 3.9.
Repeat the previous problem for \(\vec{b}=\colvec{0\\ 0}\text{,}\) \(\vec{b}=\colvec{-1\\ 2}\text{,}\) and \(\vec{b}=\colvec{2\\ 2}\text{.}\)
Question 3.20.
Can you write \(2+4t\) as a linear combination of \(1+t\) and \(-1+t\text{?}\)
Question 3.21.
Can you write \(\begin{bmatrix} 2\amp 3 \\1\amp 1 \end{bmatrix}\) as a linear combination of \(\begin{bmatrix} 1\amp 1 \\0\amp 1 \end{bmatrix}\) and \(\begin{bmatrix} 1\amp 2 \\-1\amp 1 \end{bmatrix}\text{?}\)
The span of a set of vectors \(S\text{,}\) denoted \(span(S)\) is the set of all possible linear combinations of \(S\text{.}\) A set \(S\) is said to span or generate a vector space \(V\) if \(span(S)=V\text{.}\)
Question 3.22.
If \(S=\left\{ \colvec{1\\ 1\\ 1},\colvec{1\\ -1\\ 1} \right\}\text{,}\) is \(\vec{b}=\colvec{3\\ 1\\ 3} \in span(S)\text{?}\)
Exercise 3.10.
If \(S=\left\{ \colvec{1\\ 1\\
1},\colvec{1\\ -1\\ 1} \right\}\text{,}\) is \(\vec{b}=\colvec{1\\ 2\\ 3}
\in span(S)\text{?}\)
Question 3.23.
If \(S=\left\{ \colvec{1\\ 1\\ 1},\colvec{1\\ -1\\ 1} \right\}\text{,}\) is \(\vec{b}=\colvec{0\\ 0\\ 0} \in span(S)\text{?}\)
Question 3.24.
If \(S=\left\{ \colvec{1\\ 1\\ 1},\colvec{1\\ -1\\ 1},\colvec{1\\ -1\\ -1} \right\}\text{,}\) is \(\vec{b}=\colvec{1\\ 2\\ 3} \in span(S)\text{?}\)
Question 3.25.
Is \(1-t^2\) in the span of \(\{ 3, 4+t+t^2,5-t\}\text{?}\)
Exercise 3.11.
For what value(s) of \(\alpha\) and \(\beta\) is \(\vec{p}=\colvec{\beta\\ -2\\ \alpha\\ -4}\) a solution to \(A \vec{x}=\vec{b}\) if \(A = \begin{bmatrix} 1\amp 5\amp -2\amp 0 \\ -3\amp 1\amp 9\amp -5 \\ 4\amp -8\amp -1\amp 7 \end{bmatrix}\) and \(\vec{b}=\colvec{-7\\ 9\\ 0}\text{?}\)
Question 3.26.
Is \(\vec{b}=\colvec{-7\\ 9\\ 0}\) in the span of the set of columns of \(A = \begin{bmatrix} 1\amp 5\amp -2\amp 0 \\ -3\amp 1\amp 9\amp -5 \\ 4\amp -8\amp -1\amp 7 \end{bmatrix}\text{?}\) If so, what are the coefficients?
Theorem 3.12.
Let \(V\) be a vector space. If \(S=\{\vec{v_1},\vec{v_2},\dots,\vec{v_k}\}\subseteq V\) is a set of \(k\) vectors from \(V\text{,}\) then \(span(S)\) is a subspace of \(V\text{.}\)
Question 3.27.
Find a finite set of vectors that generates each of the following vector spaces (be sure to show why your set works):
(a)
\(\mathbb{R}^3\)
(b)
\(\mathbb{P}_2\)
(c)
\(Sym_{n \times n}\)
Question 3.28.
Show that the set \(\{
1+t,t+t^2,1+t^3,t+t^2+t^3 \}\) spans all of \(\mathbb{P}_3\text{.}\)
Hint.
Come up with a system of equations that you will need to solve and use your theorems from earlier chapters.
Question 3.29.
Geometrically describe the span of \(\left\{ \colvec{2\\ 1\\ 4} \right\} \text{.}\)
Question 3.30.
Geometrically describe the span of \(\left\{ \colvec{2\\ 1\\ 4} , \colvec{3\\ -1\\ 1} \right\} \text{.}\)
Question 3.31.
Does the span of \(\left\{ \colvec{2\\ 1\\ 4} , \colvec{3\\ -1\\ 1} \right\} \) have to go through the origin?
Question 3.32.
Does the span of \(\left\{
\vec{v_1},...,\vec{v_k} \right\} \) where \(\vec{v_i} \in
\mathbb{R}^n\) have to go through the origin?
