Chapter 4 - Vector Spaces - 4.7 Change of Basis - Problems - Page 318: 6

$(-1,4,6)$

We know that $a(3,-1,-1)+b(1,-6,3)+c(0,5,-1)=(1,7,7)$. Thus $3a+b=1 \\ -a-6b+5c=7\\ -a+3b-c=7$. Thus $-a-6b+5c=7\\ -5a+15b-5c=7$ Thus $-6a+9b=42\\ 2(3a+b=1)$ Thus $11b=44 \rightarrow b=4$ Substitute: $3a+4=1 \rightarrow a=-1$ then $-(-1)-6.4+5c=7 \rightarrow c=6$ Thus the vector is $(-1,4,6)$.