Question

Express (7,0,9) as a linear combiantion of v1(1,1,1) v2(1,1,0) and v3(1,0,0)

Answer 1

Let $u=(7, 0, 9), u=av_1+bv_2+cv_3.$ Then

$$\begin{matrix} a(1)+b(1)+c(1)=7 \\ a(1)+b(1)+c(0)=0 \\ a(1)+b(0)+c(0)=9 \\ \end{matrix}$$

$$\begin{matrix} c=7 -(a+b)\\ b=-a \\ a=9 \\ \end{matrix}$$

$$\begin{matrix} c=7 \\ b=-9 \\ a=9 \\ \end{matrix}$$

$$u=(7, 0, 9)=9v_1-9v_2+7v_3$$