linear independence

A set of vectors is linearly independent when none of them can be built out of the others. Nobody is redundant: every vector points in a genuinely new direction that the rest cannot reach by scaling and adding.

There is a clean test. Ask: in how many ways can you mix the vectors to land exactly on zero? If the ONLY way is to give every vector a weight of zero, the set is independent. If some non-zero recipe also reaches zero, one vector secretly equals a combination of the others, and the set is dependent.

Example: (1,0) and (0,1) are independent (you cannot stretch one to become the other). But (1,0), (0,1), (2,3) are dependent, since (2,3) = 2*(1,0) + 3*(0,1) is already covered by the first two.