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Section 2.2 Linear Combinations (V2)

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

\begin{equation*} \left[\begin{array}{c} 3 \\ 0 \\ 5 \end{array}\right] = 2 \left[\begin{array}{c} 1 \\ -1 \\ 2 \end{array}\right] + 1\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right]\text{.} \end{equation*}
Definition 2.2.2.

The span of a set of vectors is the collection of all linear combinations of that set:

\begin{equation*} \vspan\{\vec v_1,\vec v_2,\dots,\vec v_m\} = \setBuilder{c_1\vec v_1+c_2\vec v_2+\dots+c_m\vec v_m}{ c_i\in\IR}\text{.} \end{equation*}

For example:

\begin{equation*} \vspan\setList { \left[\begin{array}{c} 1 \\ -1 \\ 2 \end{array}\right], \left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right] } = \setBuilder { a\left[\begin{array}{c} 1 \\ -1 \\ 2 \end{array}\right]+ b\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right] }{ a,b\in\IR } \end{equation*}
Activity 2.2.1.

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 2.2.2.

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.

\begin{equation*} 1\left[\begin{array}{c}1\\2\end{array}\right]+ 0\left[\begin{array}{c}-1\\1\end{array}\right]\hspace{3em} 0\left[\begin{array}{c}1\\2\end{array}\right]+ 1\left[\begin{array}{c}-1\\1\end{array}\right]\hspace{3em} 1\left[\begin{array}{c}1\\2\end{array}\right]+ 1\left[\begin{array}{c}-1\\1\end{array}\right] \end{equation*}
\begin{equation*} -2\left[\begin{array}{c}1\\2\end{array}\right]+ 1\left[\begin{array}{c}-1\\1\end{array}\right]\hspace{3em} -1\left[\begin{array}{c}1\\2\end{array}\right]+ -2\left[\begin{array}{c}-1\\1\end{array}\right] \end{equation*}
(b)

Sketch a representation of all the vectors belonging to \(\vspan\left\{\left[\begin{array}{c}1\\2\end{array}\right], \left[\begin{array}{c}-1\\1\end{array}\right]\right\}\) in the \(xy\) plane.

Activity 2.2.3.

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 2.2.4.
(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 \(\vspan\left\{\left[\begin{array}{c}1\\0\\-3\end{array}\right], \left[\begin{array}{c}-1\\-3\\2\end{array}\right]\right\}\text{?}\)

Observation 2.2.4.

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 2.2.5.

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 2.2.6.

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 2.2.7.

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)

Answer this equivalent question, and use its solution to answer the original question.

Activity 2.2.8.

Does the polynomial \(x^2+x+1\) belong to \(\vspan\{x^2-x,x+1, x^2-1\}\text{?}\)

Activity 2.2.9.

Does the matrix \(\left[\begin{array}{c}3&-2\\1&5\end{array}\right]\) belong to \(\vspan\left\{\left[\begin{array}{c}1&0\\-3&2\end{array}\right], \left[\begin{array}{c}-1&-3\\2&2\end{array}\right]\right\}\text{?}\)

(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.

Exercises 2.2.1 Exercises

1.

Consider the statement

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{ccc|c} 3 & 0 & 4 & -2 \\ 2 & 1 & -3 & -6 \\ -5 & -3 & -3 & -6 \\ -3 & 1 & 0 & -4 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement

  2. \(\left[\begin{array}{c} -2 \\ -6 \\ -6 \\ -4 \end{array}\right] \) is not a linear combination of the vectors \(\left[\begin{array}{c} 3 \\ 2 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ -3 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -3 \\ -3 \\ 0 \end{array}\right] \text{.}\)

2.

Consider the statement

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc|c} 4 & -5 & -3 & -2 & -6 \\ -2 & 0 & 1 & -2 & 1 \\ 1 & 4 & -4 & 2 & 7 \\ 0 & -3 & -1 & -2 & -2 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement

  2. \(\left[\begin{array}{c} -6 \\ 1 \\ 7 \\ -2 \end{array}\right] \) is a linear combination of the vectors \(\left[\begin{array}{c} 4 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ -4 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -2 \\ -2 \\ 2 \\ -2 \end{array}\right] \text{.}\)

3.

Consider the statement

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{ccccc|c} 2 & 4 & 8 & 12 & 0 & -7 \\ -3 & -1 & -2 & 2 & -5 & 6 \\ -3 & -3 & -6 & -6 & -3 & -1 \\ 0 & 2 & 4 & 8 & -2 & -9 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & -2 & 2 & 0 \\ 0 & 1 & 2 & 4 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement

  2. \(\left[\begin{array}{c} -7 \\ 6 \\ -1 \\ -9 \end{array}\right] \) is not a linear combination of the vectors \(\left[\begin{array}{c} 2 \\ -3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 8 \\ -2 \\ -6 \\ 4 \end{array}\right] , \left[\begin{array}{c} 12 \\ 2 \\ -6 \\ 8 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ -5 \\ -3 \\ -2 \end{array}\right] \text{.}\)

4.

Consider the statement

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc|c} 2 & -4 & -1 & -1 & 7 \\ 2 & -2 & -2 & 2 & 10 \\ 2 & -4 & -4 & -3 & 16 \\ -1 & 1 & -3 & -2 & 7 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement

  2. \(\left[\begin{array}{c} 7 \\ 10 \\ 16 \\ 7 \end{array}\right] \) is a linear combination of the vectors \(\left[\begin{array}{c} 2 \\ 2 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -4 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -1 \\ 2 \\ -3 \\ -2 \end{array}\right] \text{.}\)

5.

Consider the statement

  1. Write an equivalent statement using a vector equation.
  2. Explain why your statement is true or false.
Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{ccccc|c} -1 & 4 & -2 & -4 & -4 & -5 \\ -1 & -2 & -2 & 8 & 2 & -4 \\ -3 & -5 & -6 & 22 & 5 & 2 \\ 1 & -1 & 2 & -2 & 1 & 3 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 2 & -4 & 0 & 0 \\ 0 & 1 & 0 & -2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
  1. The statement is equivalent to the statement

  2. \(\left[\begin{array}{c} -5 \\ -4 \\ 2 \\ 3 \end{array}\right] \) is not a linear combination of the vectors \(\left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -6 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 8 \\ 22 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ 2 \\ 5 \\ 1 \end{array}\right] \text{.}\)

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