91Ó°ÊÓ

\({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right]\)and \({v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\)

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\(c\).

The linear combination of two vectors\({{\mathop{\rm v}\nolimits} _1}\)and\({{\mathop{\rm v}\nolimits} _2}\)is a multiple of\({{\mathop{\rm v}\nolimits} _1}\). Span\(\left\{ {{v_1},...,{v_p}} \right\}\)contains every scalar multiple of\({{\mathop{\rm v}\nolimits} _1}\).

Write the vectors\({{\mathop{\rm v}\nolimits} _1}\)and\({{\mathop{\rm v}\nolimits} _2}\)in a linear combination

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

Suppose\({\mathop{\rm v}\nolimits} \)is a nonzero vector in\({\mathbb{R}^3}\), then, span\(\left\{ {\mathop{\rm v}\nolimits} \right\}\)is a set of all scalar multiples of\({\mathop{\rm v}\nolimits} \), which is the set of points on the line in\({\mathbb{R}^3}\)through\({\mathop{\rm v}\nolimits} \)and 0. If\({\mathop{\rm u}\nolimits} \)and\(v\)are nonzero vectors in\({\mathbb{R}^3}\), where\({\mathop{\rm v}\nolimits} \)is not a multiple of\({\mathop{\rm u}\nolimits} \), then span\(\left\{ {u,v} \right\}\)is the plane in\({\mathbb{R}^3}\)that contains\({\mathop{\rm u}\nolimits} ,v\)and origin 0. In particular, span\(\left\{ {{\rm{u,v}}} \right\}\)contains the line in\({\mathbb{R}^3}\)through\({\mathop{\rm u}\nolimits} \)and 0 and the line through \({\mathop{\rm v}\nolimits} \)and 0.

Vector \({{\mathop{\rm v}\nolimits} _2}\)is not a scalar multiple of \({{\mathop{\rm v}\nolimits} _1}\).

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

Draw the graph of the geometric description of a span \({\mathbb{R}^3}\)

Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{v_2}} \right\}\) is a plane in \({\mathbb{R}^3}\) through the origin since the vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) are not a multiple of each other. In conventional 3-space, every vector in the set has 0 as its second entry, and therefore resides in the xz-plane.

Therefore, span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{v_2}} \right\}\) is a plane in \({\mathbb{R}^3}\) through the origin since the vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) are not a multiple of each other.