Give a geometric description of span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right]\) and \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{12}\\3\\{ - 9}\end{array}} \right]\).

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

\(a{v_1} + b{v_2}\)

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\({\mathbb{R}^3}\), 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.

Write vector \({{\mathop{\rm v}\nolimits} _2}\) as an expression of \({{\mathop{\rm v}\nolimits} _1}\) in the linear combination of vectors

\(\begin{array}{l}a{v_1} + b{v_2} = a\left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right] + b\left[ {\begin{array}{*{20}{c}}{12}\\3\\{ - 9}\end{array}} \right]\\a{v_1} + b{v_2} = a\left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right] + b\left( {\frac{3}{2}} \right)\left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right]\\a{v_1} + b{v_2} = a{v_1} + \frac{{3b}}{2}{v_1}\\a{v_1} + b{v_2} = \left( {a + \frac{{3b}}{2}} \right){v_1}\end{array}\)

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

Since \({{\mathop{\rm v}\nolimits} _2}\) is a multiple of \({{\mathop{\rm v}\nolimits} _1}\), span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{v_2}} \right\}\) is a line in \({\mathbb{R}^3}\) spanned by \({v_1}\) and \({{\mathop{\rm v}\nolimits} _2}\).

Therefore, span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{v_2}} \right\}\) is the pair of points on the line passing through \({v_1}\) and 0.