Discover vectors as arrows, not just lists of numbers. Explore addition, scaling, and what it means to span a space.
Before diving into matrices, let's build intuition for vectors. A vector isn't just a list of numbers—it's an arrow pointing from the origin to some point in space. The numbers simply tell us where the tip lands.
This geometric view is the key to understanding all of linear algebra. Every operation we'll learn has a beautiful visual meaning.
To add two vectors, place the tail of the second at the tip of the first. The sum is the vector from the original origin to the new tip—forming a parallelogram.
Drag the blue and green vector tips. Notice how the sum always completes the parallelogram.
Multiplying a vector by a number (scalar) stretches or compresses it. Positive scalars preserve direction, negative scalars flip it.
Positive scalars: stretch or compress the vector
Negative scalars: flip direction and scale
Zero: collapses the vector to a point
A linear combination takes multiple vectors, scales each by some coefficient, and adds the results. This is the fundamental operation of linear algebra.
A linear combination scales each basis vector and adds them together. Every point in the plane can be reached by some combination of v₁ and v₂.
The span of a set of vectors is all points reachable via linear combinations. One vector spans a line. Two non-parallel vectors span a plane. What happens if they're parallel?
Two non-parallel vectors span the entire 2D plane
The span of a set of vectors is all possible linear combinations. Drag the vectors to see how the span changes.
Everything extends naturally to three dimensions. Three independent vectors span all of 3D space—any point can be reached by some linear combination.
Three independent vectors span all of 3D space. Drag to rotate the view, scroll to zoom. Any point in space can be reached by some linear combination of v₁, v₂, and v₃.
Next up: We'll see how matrices transform entire spaces at once.