Question

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

15. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\)

Short answer

The five vectors in span \(\left\{ {{v_1},{v_2}} \right\}\) are \(\left\{ {\left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right],\left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\4\\{ - 6}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{12}\\{ - 2}\\{ - 6}\end{array}} \right]} \right\}\).

Step-by-step solution

Choose the five sets of weights to generate the vector

If vectors \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\) in \({\mathbb{R}^n}\) are given with scalars \({c_1},{c_2},...,{c_p}\), then the vector \({\mathop{\rm y}\nolimits} \) defined by \(y = {c_1}{v_1} + .... + {c_p}{v_p}\) is called alinear combination of\({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\)with weights\({c_1},{c_2},...,{c_p}\).

Choose the five sets of weights as:

\({\rm{w}} = \left\{ {0,0} \right\},\left\{ {1,0} \right\},\left\{ {0,1} \right\},\left\{ {1,1} \right\},\left\{ {1, - 1} \right\}\)

Generate the weight of the first vector

In\({\mathbb{R}^2}\), the sum of two vectors\({\mathop{\rm u}\nolimits} \)and\({\mathop{\rm v}\nolimits} \)is thevector addition\({\mathop{\rm u}\nolimits} + v\), which is obtained by adding the corresponding entries of \({\mathop{\rm u}\nolimits} \) and \({\mathop{\rm v}\nolimits} \).

Thescalar multiple of a vector\({\mathop{\rm u}\nolimits} \)by real number\(c\)is the vector\(c{\mathop{\rm u}\nolimits} \)obtained by multiplying each entry in\({\mathop{\rm u}\nolimits} \)by\(c\).

Use scalar multiplication and vector addition to generate the weight of the vector

\(\begin{aligned}{c}{V_1} &= 0{v_1} + 0{v_2}\\ &= 0\left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right] + 0\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\end{aligned}\)

Generate the weight of the second vector

If \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\) in \({\mathbb{R}^n}\), then the set of all linear combinations \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\) is denoted by span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}} \right\}\) and is called the subset of \({\mathbb{R}^n}\) spanned \({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\). Span is the collection of all vectors that can be written in the form \({c_1}{v_1} + {c_2}{v_2} + .... + {c_p}{v_p}\) with \({c_1},...,{c_p}\) scalars.

Use scalar multiplication and vector addition to generate the weight of the vector.

\(\begin{aligned}{c}{V_2} &= 1{v_1} + 0{v_2}\\ &= 1\left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right] + 0\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right]\end{aligned}\)

Generate the weight of the third vector

Use scalar multiplication and vector addition to generate the weight of the vector.

\(\begin{aligned}{c}{V_3} &= 0{v_1} + 1{v_2}\\ &= 0\left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right] + 1\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\end{aligned}\)

Generate the weight of the fourth vector

Use scalar multiplication and vector addition to generate the weight of the vector.

\(\begin{aligned}{c}{V_4} &= 1{v_1} + 1{v_2}\\ &= 1\left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right] + 1\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{7 - 5}\\{1 + 3}\\{ - 6 + 0}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}2\\4\\{ - 6}\end{array}} \right]\end{aligned}\)

Generate the weight of the fifth vector

Use scalar multiplication and vector addition to generate the weight of the vector.

\(\begin{aligned}{c}{V_5} &= 1{v_1} - 1{v_2}\\ &= 1\left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right] - 1\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{7 + 5}\\{1 - 3}\\{ - 6 - 0}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{12}\\{ - 2}\\{ - 6}\end{array}} \right]\end{aligned}\)

List the three entries of the vector

The three entries of the vector are \(\left\{ {\left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{12}\\{ - 2}\\{ - 6}\end{array}} \right]} \right\}\)

Hence, the five vectors in span \(\left\{ {{v_1},{v_2}} \right\}\) are \(\left\{ {\left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right],\left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\4\\{ - 6}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{12}\\{ - 2}\\{ - 6}\end{array}} \right]} \right\}\).