Question

I'm having trouble understanding an example from my book. It is trying to establish whether or not three vectors are linearly dependent. The step where I'm stuck is in the Gaussian elimination (image attached). How do they decide that c=1, and the equations that follow?

Answer 1

c can be any value. But c=0 leads to a and b also being 0, and you get the "trivial" solution <0,0,0>
the next simplest number is c=1, and this leads to b= -3 and a=4, and the non-trivial solution <4, -3, 1>
if we used c=2, we would get <8, -6, 2>
which can be written as 2<4,-3,1> i.e. the first solution scaled by 2

in general, if <a,b,c> is in the null set, then D*<a,b,c> (where D is a scale factor) is also.

so, just like in fractions, people tend to write the "simplest" vector (scale factored out of each component)

Answer 2

Ah, ok. They certainly could have done a better job justifying that step, but I understood you perfectly. Thanks! :)