Span, Linear Combinations & Basis Vectors
Two directions, scaled and stacked, can reach anywhere on a flat plane. That entire reachable set has a name: the span.
Intuition First
Think of a linear combination as a recipe for reaching any point using two fixed directions. Given two arrows to start from, you stretch or shrink each one (scaling) and then walk them one after another (adding). Change the amounts, and you land somewhere new.
These two starting directions are called basis vectors — the minimal set of arrows you need so that every reachable point can be written as some recipe of the two. The complete set of every point you could ever reach this way is called the span of those vectors.
Start With a Number You Can Check by Hand
Take the two simplest possible directions: one step right, i = (1, 0), and one step up, j = (0, 1). To reach the point (3, 2), the recipe is "3 of i, plus 2 of j":
That's a linear combination in action — every point on the plane is reachable this way, just by changing the two scaling numbers. That's exactly what the sliders below let you do.
Try It — Build Any Point From Two Directions
Move the sliders to change how much of each basis vector you use. Watch the resultant point trace out the full plane.
Formal Definition
Given a set of vectors v₁, v₂, …, vₖ and scalar numbers c₁, c₂, …, cₖ, a linear combination is:
The span is simply the name for every point you can reach this way — every possible recipe of scaling and adding those vectors:
A set of vectors forms a basis for a space if they span it using the fewest vectors possible — remove any one and some points become unreachable.