A vector is redundant if you could have built it from the others. The vectors \vec{u} and 2\vec{u} carry the same information — the second is no help, because it's already in the span of the first.

A set of vectors is linearly independent when none of them is a linear combination of the others — every vector pulls in a genuinely new direction. If even one is built from the rest, the set is linearly dependent. The crisp test: the only way to combine them to the zero vector is the boring way, with all weights zero.

New direction, or not?

Swing \vec{v} around. Whenever it lines up with \vec{u} (same or opposite heading), the two become dependent — \vec{v} is just a scaling of \vec{u}. Every other angle is independent: two directions, no redundancy, enough to span the plane.

How many can be independent?

In the plane you can have at most two independent vectors — a third must lie in their span, so it's always redundant. In 3D the limit is three. That ceiling is no accident: it is the dimension of the space, and it is the same number whichever independent vectors you happen to pick. Independence plus spanning is the winning combination we name next.