Prove that there is a no solution to a given vector equation

by RoRi

April 28, 2016

Let

$A = (2,-1,1), \qquad B = (1,2,-1), \qquad C = (2,11,7).$

Prove that there are no scalars $x,y \in \mathbb{R}$ such that $C = xA + yB$ .

Proof. If $C = xA + yB$ then we must have $x$ and $y$ satisfy the three equations

$\begin{align*} 2x + y &= 2 \\ -x + 2y &= 11 \\ x - y &= 7. \end{align*}$

From the third equation we have $x = y + 7$ . Plugging this into the second equation we have $-y-7+2y = 11$ which implies $y = 18$ . Therefore $x = 25$ . But then, the first equation is false since

$2(25) + 18 = 68 \neq 2.$

Therefore, there are no $x$ and $y$ that satisfy the vector equation $. \qquad \blacksquare$

Stumbling Robot

Prove that there is a no solution to a given vector equation

Related

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): Cancel reply