TITLE: Intuition for Formal Definition of Linear Independence
QUESTION [12 upvotes]: I learned about this a long time ago but it never really clicked, which led me to these questions:

How the formal definition (at the bottom) works. I have a rough intuition: linear independence is where the variables are independent and don't affect each other. But I don't follow the formal definition. I would like to have a deep understanding of the formal definition based on these linear combination equations. I'm not sure how a linear combination constructed in a certain way can tell you the variables are independent or not.
Why set the linear combination equations to $\vec{0}$. I don't see how setting to zero helps determine independence or not.
Why choose $a_i$ to be non-zero in one case. It seems arbitrary.


From Wikipedia:

A subset $S=\{{\vec {v}}_{1},{\vec {v}}_{2},\dots ,{\vec {v}}_{n}\}$ of a vector space $V$ is linearly dependent if there exist a finite number of distinct vectors ${\vec {v}}_{1},{\vec {v}}_{2},\dots ,{\vec {v}}_{k}$ in $S$ and scalars $a_{1},a_{2},\dots ,a_{k}$, not all zero, such that
$$a_{1}{\vec {v}}_{1}+a_{2}{\vec {v}}_{2}+\cdots +a_{k}{\vec {v}}_{k}={\vec {0}}$$
where ${\vec {0}}$ denotes the zero vector.
The vectors in a set $T=\{{\vec {v}}_{1},{\vec {v}}_{2},\dots ,{\vec {v}}_{n}\}$ are linearly independent if the equation
$$a_{1}{\vec {v}}_{1}+a_{2}{\vec {v}}_{2}+\cdots +a_{n}{\vec {v}}_{n}={\vec {0}}$$
can only be satisfied by $a_{i}=0$ for $i=1,\dots ,n$.

So my understanding is, there are two subsets $S$ and $T$ of $V$. In one of them, the coefficients are not all zero, in the other they are all zero. In one case they are linearly dependent, in the other not. I don't understand why though; that's as much as I understand. Not sure why the equations were constructed like this in the first place.

REPLY [3 votes]: Your intuition may be leading you astray. You write "linear independence is where the variables are independent and don't affect each other." But vectors are not variables. $(1, 0)$, $(0, 1)$, and $(1, 1)$ are linearly dependent, but that's because you can get $(1, 1)$ by adding the first two vectors together. It's not because changing one of the vectors will change the other, since they are what they are. 
The idea of one vector being dependent on others is that you can get the first vector by scaling the other vectors appropriately and then adding the scaled vectors together. For instance, $(2, 4)$ is dependent on $(1, 0)$ and $(0, 1)$ because 
$$
2 \cdot (1, 0) + 4 \cdot (0, 1) = (2, 4)
$$
Since you can represent $(2, 4)$ by multiplying $(1,0)$ and $(0,1)$ by constants and adding the results together, it is not independent of them. 
But you could take the equation and move the $(2,4)$ to the other side, like this:
$$
2 \cdot (1, 0) + 4 \cdot (0, 1) + (-1) \cdot(2,4) = 0
$$
And that's an example of just what the definition is saying. When you say that you can get a sum like that to add up to zero, it's the same as saying that you can get any one of the vectors by multiplying the others by appropriate constants and adding them. And that's what it means to not be independent.