how do you find out if a vector is linearly dependent or not?

Question

anonymous · Answer

so for vectors v1,v2,v3,v4,........vn  there exists scalars s1,s2,s3,s4......sn
such that 
s1*v1 +s2*v2+  s3*v3 +.......................................sn*vn=0
then vectors v1,v2 ..... vn are linearly independent if each of s1,s22,s3...sn is 0
and linearly dependent if at least one of  s1,s22,s3...sn is not 0

anonymous · Answer

soo if my vectors are [0 2 3], [0 0 -8], and [1 -3 1], they would be independent?

kinggeorge · Answer

I believe those would be independent. Suppose that there are some scalars $c_1,c_2\in\mathbb{R}$ such that $$c_1\cdot[0,2,3]+c_2[0,0,8]=[1,-3,1]$$So$$[0,2c_1,3c_1]+[0,0,8c_2]=[1,-3,1]$$However, $0+0=0\neq1$ for any choice of $c_1$ and $c_2$. You can make similar arguments to show that $[0,2,3]$ and $[0,0,8]$ are similarly independent. This would show that all three are linearly independent.

anonymous · Answer

woahhh that makes so much sense! [: thank you :) can you make that argument all the time?

kinggeorge · Answer

You can try to make that argument. If you're working in $\mathbb{R}$ or the complex numbers, it's usually fairly straightforward (at least in what you will be expected to do). 

An alternative method, which works particularly well when you have 3 or 2-dimensional vectors, is to put them in a matrix. $$\begin{bmatrix}1&-3&1\0&2&3\0&0&8\end{bmatrix}$$If you can row-reduce to this form, they're linearly independent.

kinggeorge · Answer

Another thing to watch for, is the dimension of the vector space. If you have an $n$ dimensional vector space, and $m$ vectors, where $m>n$, then your vectors are not linearly independent.