91Ó°ÊÓ

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:

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

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 multipleof 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

\(\begin{aligned}{c}{V_1} &= 0{v_1} + 0{v_2}\\ &= 0\left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right] + 0\left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\end{aligned}\)

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 \({c_1}{v_1} + {c_2}{v_2} + .... + {c_p}{v_p}\) of \({c_1},...,{c_p}\) scalars.

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

\(\begin{aligned}{c}{V_2} &= 1{v_1} + 0{v_2}\\ &= 1\left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right] + 0\left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{3 + 0}\\{0 + 0}\\{2 + 0}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right]\end{aligned}\)

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

\(\begin{aligned}{c}{V_3} &= 0{v_1} + 1{v_2}\\ &= 0\left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right] + 1\left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\end{aligned}\)

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

\(\begin{aligned}{c}{v_4} &= 1{v_1} + 1{v_2}\\ &= 1\left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right] + 1\left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{3 - 2}\\{0 + 0}\\{2 + 3}\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}1\\0\\5\end{array}} \right]\end{aligned}\)

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

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

The weights on \({v_1}\) and \({v_2}\) to generate the vector are \(\left\{ {0,0} \right\},\left\{ {1,0} \right\},\left\{ {2,3} \right\},\left\{ {1,1} \right\},\left\{ {1, - 1} \right\}\)

The three entries of the vector are \(\left\{ {\left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right],\left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right],\left[ {\begin{array}{*{20}{c}}0\\0\\{13}\end{array}} \right]} \right\}\)

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