# Understanding linear combinations and independence

• Mar 26th 2009, 01:27 AM
Craka
Understanding linear combinations and independence
Guys, I've looked in a few places to try and get an understanding of this, and I'm pretty confused to be perfectly honest.

From what I can understand for vectors to be linear dependent, they have to be able to be put in a linear combination.

I'm having trouble how to work out how to put vectors into a linear combination.
For instance on wikipedia, http://en.wikipedia.org/wiki/Linearly

$\displaystyle \left( {\begin{array}{*{20}c} 0 & 0 & 1 & 4 \\ 0 & 2 & { - 2} & 2 \\ 1 & { - 2} & 1 & 3 \\ \end{array}} \right)$

They show 3 vectors as being independent, but the total of 4 vectors being dependent, I can understand the 4 vector case as in R^3 can only have 3 vectors being independent of each other. However I don't understand how the determine that the first 3 vectors are independent

They show a linear combination of
$\displaystyle v_1 = \left( {\frac{{ - 5}}{9}} \right)v_2 + \left( {\frac{{ - 4}}{9}} \right)v_3 + \left( {\frac{1}{9}} \right)v_4$

They give a description of why this is a linear combination but I don't seem to follow that either. Would be very grateful for a explanation of the going on here. Thanks
• Mar 26th 2009, 04:03 AM
Showcase_22
They're using simultaneous equations.

$\displaystyle v_1=\alpha_2v_2+\alpha_3v_3+\alpha_4v_4$ and solving for each of the $\displaystyle \alpha_i$'s.