There is a three-dimensional vector $v$. Show that $v$ can be expressed as a linear combination of $v_1$, $v_2$, and $v_3$, where $v_1 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$, $\quad v_2 = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}$, $\quad v_3 = \begin{pmatrix} 11 \\ 1 \\ -14 \end{pmatrix}$.
I know that I am supposed to use eigenvalues ($Mv = kv$) and linear combinations of vectors, like in here: http://www.vitutor.com/geometry/vectors/linear_combination.html, but I'm not sure how to use these terms to solve this problem, or even begin it.
Best Answer
Hint
Let $v=\begin{pmatrix}a\\b\\c\end{pmatrix}$. You need to see if there exists $x,y,z$ (scalars) such that $$v=xv_1+yv_2+zv_3.$$ In other words you need to see if the following system is consistent for all $a,b,c$. $$ \left[\begin{array}{ccc|l} 1 & -1 & 11 &a\\ 1 & 1 & 1 & b\\ 1 & 1 & -14 &c\\ \end{array} \right] $$