Question

For Which vlue of K does the vector (1,2,k) belong to the linear span of [{(1,1,1),(2,1,1)}

Answer 1

Perform elimination on the following system:
\[\left[\begin{matrix}1 & 2 \\ 1 & 1\\ 1 & 1 \end{matrix}\right]\left(\begin{matrix}c \\ d\end{matrix}\right)=\left(\begin{matrix}1 \\ 2 \\ k\end{matrix}\right)\]

to arrive at a 3 by 3 system with row 3:
\[\left[\begin{matrix}1 & +2 | 1\\ 0 & -1|1\\ 0 &0|k-2\end{matrix}\right]\]

In order for this system of equations to be consistent i.e. to have a solution, then k-2 = 0 => k=2. You will find that c=3 and d=-1 which is correct as well if plugged into the original system.

Answer 2

consider the linear span of \(\{(1,1,1),(2,1,1)\}\) gives vectors $$a(1,1,1)+b(2,1,1)=(a+2b,a+b,a+b)$$

Answer 3

hence we have the system:$$a+2b=1\\a+b=2\\a+b=k$$so clearly \(k=2\)