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.
Best Answer
Imagine you have a collection of arrows pointing in various directions. If they're linearly dependent, then you can stretch, shrink, and reverse (but not rotate) them in such a way that if you lay them head-to-tail then they form a closed loop. For example, if you have three arrows that happen to all lie in the same plane (linearly dependent), then you can form a triangle out of them, but you can't if one of them sticks out of the plane formed by the other two (linearly independent).