Section 2.2 Linear Combinations (VS2)
Learning Outcomes
Determine if a Euclidean vector can be written as a linear combination of a given set of Euclidean vectors by solving an appropriate vector equation.
Subsection 2.2.1 Class Activities
Definition 2.2.1.
A linear combination of a set of vectors \(\{\vec v_1,\vec v_2,\dots,\vec v_m\}\) is given by \(c_1\vec v_1+c_2\vec v_2+\dots+c_m\vec v_m\) for any choice of scalar multiples \(c_1,c_2,\dots,c_m\text{.}\)
For example, we can say \(\left[\begin{array}{c}3 \\0 \\ 5\end{array}\right]\) is a linear combination of the vectors \(\left[\begin{array}{c} 1 \\ -1 \\ 2 \end{array}\right]\) and \(\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right]\) since
Definition 2.2.2.
The span of a set of vectors is the collection of all linear combinations of that set:
For example:
Activity 2.2.3.
Consider \(\vspan\left\{\left[\begin{array}{c}1\\2\end{array}\right]\right\}\text{.}\)
(a)
Sketch \(1\left[\begin{array}{c}1\\2\end{array}\right]=\left[\begin{array}{c}1\\2\end{array}\right]\text{,}\) \(3\left[\begin{array}{c}1\\2\end{array}\right]=\left[\begin{array}{c}3\\6\end{array}\right]\text{,}\) \(0\left[\begin{array}{c}1\\2\end{array}\right]=\left[\begin{array}{c}0\\0\end{array}\right]\text{,}\) and \(-2\left[\begin{array}{c}1\\2\end{array}\right]=\left[\begin{array}{c}-2\\-4\end{array}\right]\) in the \(xy\) plane.
(b)
Sketch a representation of all the vectors belonging to \(\vspan\setList{\left[\begin{array}{c}1\\2\end{array}\right]} = \setBuilder{a\left[\begin{array}{c}1\\2\end{array}\right]}{a\in\IR}\) in the \(xy\) plane.
Activity 2.2.4.
Consider \(\vspan\left\{\left[\begin{array}{c}1\\2\end{array}\right], \left[\begin{array}{c}-1\\1\end{array}\right]\right\}\text{.}\)
(a)
Sketch the following linear combinations in the \(xy\) plane.
(b)
Sketch a representation of all the vectors belonging to \(\vspan\left\{\left[\begin{array}{c}1\\2\end{array}\right], \left[\begin{array}{c}-1\\1\end{array}\right]\right\}=\setBuilder{a\left[\begin{array}{c}1\\2\end{array}\right]+b\left[\begin{array}{c}-1\\1\end{array}\right]}{a, b \in \IR}\) in the \(xy\) plane.
Activity 2.2.5.
Sketch a representation of all the vectors belonging to \(\vspan\left\{\left[\begin{array}{c}6\\-4\end{array}\right], \left[\begin{array}{c}-3\\2\end{array}\right]\right\}\) in the \(xy\) plane.
Activity 2.2.6.
The vector \(\left[\begin{array}{c}-1\\-6\\1\end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\0\\-3\end{array}\right], \left[\begin{array}{c}-1\\-3\\2\end{array}\right]\right\}\) exactly when there exists a solution to the vector equation \(x_1\left[\begin{array}{c}1\\0\\-3\end{array}\right]+ x_2\left[\begin{array}{c}-1\\-3\\2\end{array}\right] =\left[\begin{array}{c}-1\\-6\\1\end{array}\right]\text{.}\)
(a)
Reinterpret this vector equation as a system of linear equations.
(b)
Find its solution set, using technology to find \(\RREF\) of its corresponding augmented matrix.
(c)
Given this solution set, does \(\left[\begin{array}{c}-1\\-6\\1\end{array}\right]\) belong to \(\vspan\left\{\left[\begin{array}{c}1\\0\\-3\end{array}\right], \left[\begin{array}{c}-1\\-3\\2\end{array}\right]\right\}\text{?}\)
Fact 2.2.7.
A vector \(\vec b\) belongs to \(\vspan\{\vec v_1,\dots,\vec v_n\}\) if and only if the vector equation \(x_1 \vec{v}_1+\cdots+x_n \vec{v}_n=\vec{b}\) is consistent.
Observation 2.2.8.
The following are all equivalent statements:
The vector \(\vec{b}\) belongs to \(\vspan\{\vec v_1,\dots,\vec v_n\}\text{.}\)
The vector equation \(x_1 \vec{v}_1+\cdots+x_n \vec{v}_n=\vec{b}\) is consistent.
The linear system corresponding to \(\left[\vec v_1\,\dots\,\vec v_n \,|\, \vec b\right]\) is consistent.
\(\RREF\left[\vec v_1\,\dots\,\vec v_n \,|\, \vec b\right]\) doesn't have a row \([0\,\cdots\,0\,|\,1]\) representing the contradiction \(0=1\text{.}\)
Activity 2.2.9.
Determine if \(\left[\begin{array}{c}3\\-2\\1 \\ 5\end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\0\\-3 \\ 2\end{array}\right], \left[\begin{array}{c}-1\\-3\\2 \\ 2\end{array}\right]\right\}\) by solving an appropriate vector equation.
Activity 2.2.10.
Determine if \(\left[\begin{array}{c}-1\\-9\\0\end{array}\right]\) belongs to \(\vspan\left\{\left[\begin{array}{c}1\\0\\-3\end{array}\right], \left[\begin{array}{c}-1\\-3\\2\end{array}\right]\right\}\) by solving an appropriate vector equation.
Activity 2.2.11.
Does the third-degree polynomial \(3y^3-2y^2+y+5\) in \(\P_3\) belong to \(\vspan\{y^3-3y+2,-y^3-3y^2+2y+2\}\text{?}\)
(a)
Reinterpret this question as a question about the solution(s) of a polynomial equation:
(b)
Write a Euclidean vector equation that has the same solution set:
(c)
Answer this equivalent question, and use its solution to answer the original question.
Activity 2.2.12.
Does the polynomial \(x^2+x+1\) belong to \(\vspan\{x^2-x,x+1, x^2-1\}\text{?}\)
Activity 2.2.13.
Does the matrix \(\left[\begin{array}{cc}3&-2\\1&5\end{array}\right]\) belong to \(\vspan\left\{\left[\begin{array}{cc}1&0\\-3&2\end{array}\right], \left[\begin{array}{cc}-1&-3\\2&2\end{array}\right]\right\}\text{?}\)
(a)
Reinterpret this question as a question about the solution(s) of a matrix equation.
(b)
Answer this equivalent question, and use its solution to answer the original question.
Subsection 2.2.2 Videos
Subsection 2.2.3 Slideshow
Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/VS2.slides.html
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Exercises 2.2.4 Exercises
Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/VS2/
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Subsection 2.2.5 Mathematical Writing Explorations
Exploration 2.2.14.
Suppose \(S = \{\vec{v_1},\ldots, \vec{v_n}\}\) is a set of vectors. Show that \(\vec{v_0}\) is a linear combination of members of \(S\text{,}\) if an only if there are a set of scalars \(\{c_0,c_1,\ldots, c_n\}\) such that \(\vec{z} = c_0\vec{v_0} + \cdots + c_n\vec{v_n}.\) We can do this in a few parts. I've used bullets here to indicate all that needs to be done. This is an "if and only if" proof, so it needs two parts.
First, assume that \(\vec{0} = c_0\vec{v_0} + \cdots + c_n\vec{v_n}\) has a solution, with \(c_0 \neq 0\text{.}\) Show that \(\vec{v_0}\) is a linear combination of elements of \(S\text{.}\)
Next, assume that \(\vec{v_0}\) is a linear combination of elements of \(S\text{.}\) Can you find the appropriate \(\{c_0,c_1,\ldots, c_n\}\) to make the equation \(\vec{z} = c_0\vec{v_0} + \cdots + c_n\vec{v_n}\) true?
In either of your proofs above, does the case when \(\vec{v_0} = \vec{z}\) change your thinking? Explain why or why not.
Subsection 2.2.6 Sample Problem and Solution
Sample problem Example B.1.6.