Let $x_1,x_2,\dots,x_6$ be variables.
Let $c_1,c_2,c_3,c_4>0$ be positive real numbers.
Consider the four expressions:
- $w_2:= c_2x_6-c_3x_5+c_4x_4$
Something nice is that:
(we can tediously verify it)
I.e. there exists a nontrivial linear combination of $w_1,w_2,w_3,w_4$ that results in zero.
My question is what is the underlying principle behind this? (the existence of a nontrivial linear combination that results in zero.) Is there any reason from linear algebra or others?
Or is it due to some pattern of the $w_1,w_2,w_3,w_4$?
Hope this question makes sense. Thanks.
Another way to phrase my question is how do we get this relation $$c_4w_1-c_1w_2-c_3w_3+c_2w_4=0$$ (or any other linear combination that results in zero) without doing guessing or complicated solving of system of equations? And in the first place, how do we know if such a relation exists?
One way is to try to match one variable by one variable, say consider $x_4$. Then the hint is to multiply $w_1$ by $c_4$ and $w_2$ by $-c_1$ to "eliminate" $x_4$. But why does this method work out so nicely?
Thanks once again.