Chapter 4 - Vector Spaces - 4.7 Change of Basis - Problems - Page 318: 4

$(-4,1,3)$

We know that $a(1,0,1)+b(1,1,-1)+c(2,0,1)=(-9,1,-8)$. Thus $a+b+2c=-9 \\b=1$ and $a-b+c=-8$. Thus $a+1+2c=-9\\b=1\\a-1+c=-8$. Thus $a+2c=-10\\b=1\\a+c=-7$ Thus $c=-3 \\b=1 \\ a+c=-7$ Thus $c=-3 \\ b=1 \\ a=-4$ Thus the vector is $(-4,1,3)$.