Linear Algebra, Independent Vectors
Independent Vectors
A set of vectors is linearly independent if there is no linear combination of these vectors
that produces 0, except for the trivial linear combination, where all coefficients are 0.
By "linear combination", we mean a finite sum of scaled vectors.
This is not the time to build an infinite series.
In our example, [1,1] and [-1,1] are independent.
They point in different directions;
you can't add two scaled versions of these vectors and get the origin.
Algebraically, assume s×[1,1] + t×[-1,1] = [0,0],
hence s+t = s-t = 0, and s and t must be 0.
Note that [2,2] and [-3,-3] are not independent, since 3 times the first + 2 times the second is 0.