Originally Posted by

**topsquark** Here we need to define linearly dependent a bit more carefully. If three vectors are linearly dependent we may take a linear combination of any two of them and produce a third. So one example of this is

$\displaystyle \left[ \begin {matrix}a+4 \\ 3

\\ 2 \end {matrix} \right] = c \left[ \begin {matrix} a

\\ 1 \\ 0 \end {matrix} \right] +

d \left[ \begin {matrix} 2\\ 1 \\ 1

\end {matrix} \right] $

where c and d are some constants.

So we have the equations:

$\displaystyle a + 4 = c \cdot a + d \cdot 2$

$\displaystyle 3 = c \cdot 1 + d \cdot 1$

$\displaystyle 2 = c \cdot 0 + d \cdot 1$

I'll let you solve the system for a, c, and d.

Note: You may set up the 2 other examples, reordering the vectors as you do, and you will get the same value for a, but obviously with different values of c and d for each.

-Dan