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

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

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