Answer to Question #224526 in Linear Algebra for Anuj

Question

Let u=(2,6,-7), v=(-1,-1,8), and 3. If (2, 14, 11)=ku+lv what is the value of l?

Answer 1

$\textsf{Given:}\\ u=(2,6,-7)\\ v=(-1,-1,8)\\ ku+lv= (2, 14, 11)----------(i)\\ ku=k(2,6,-7)=(2k,6k,-7k)\\ lv=l(-1,-1,8)=(-l,-l,8l)\\ ku+lv=(2k-l,6k-l,-7k+8l)---(ii)\\ \textsf{Comparing }(i)\textsf{ and }(ii),\\ 2k-l=2\\ 6k-l=14\\ -7k+8l=11\\ \textsf{Hence,}\\ 2k-l-2=0------------(iii)\\ 6k-l-14=0-----------(iv)\\ -7k+8l-11=0----------(v)\\ \textsf{By solving either (iii) and (iv), (iv) and (v),} \\ \textsf{or (iii) and (v) simultaneously, we obtain}\\ k=-3, l=-4.\\ \textsf{Hence, the value of } l=-4.$