Chapter 11.2, Problem 58ES

A vector w is a linear combination of the vectors ${\text{v}}_1,{\text{v}}_2,\text{and}{\text{v}}_3$ if w can be expressed as $\text{w}={\text{c}}_1{\text{v}}_1+{\text{c}}_2{\text{v}}_2+{\text{c}}_3{\text{v}}_3,$ Where $c_1,c_2,\text{and}{c}_3$ are scalars.

(a) Find scalars $c_1,c_2,\text{and}{c}_3$ to express $〈-1,1,5〉$ as a linear combination of ${\text{v}}_1=〈1,0,1〉,{\text{v}}_2=〈3,2,0〉,$ and ${\text{v}}_3=〈0,1,1〉.$

(b) Show that the vector $2\text{i}+\text{j}-\text{k}$ cannot be expressed as a linear combination of ${\text{v}}_{\text{1}}=\text{i}-\text{j},{\text{v}}_2=3\text{i}+\text{k},\text{and}{\text{v}}_3=4\text{i}-\text{j}+\text{k}\text{.}$