Vector Space
A set of vectors that can be added together and multiplied by scalars (numbers) while satisfying certain rules. In this visualizer, we work with R² (2D space) and R³ (3D space).
Linear Combination
An expression constructed from a set of vectors by multiplying each vector by a scalar and adding the results. For vectors V₁, V₂, ..., Vₙ and scalars C₁, C₂, ..., Cₙ:
C₁V₁ + C₂V₂ + ... + CₙVₙ
This creates a new vector that lies within the span of the original vectors.
Span
The set of all possible linear combinations of a set of vectors. It represents all the vectors that can be "reached" using the given vectors.
- One vector spans a line
- Two independent vectors span a plane
- Three independent vectors in R³ span the entire 3D space
Linear Independence
A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. In other words, each vector adds a new "direction" that wasn't already covered.
If vectors are linearly dependent, at least one vector is redundant and doesn't contribute to expanding the span.
Basis
A set of vectors that is both linearly independent and spans the entire vector space. A basis provides the minimal set of vectors needed to describe every vector in the space.
In R², any two independent vectors form a basis. In R³, any three independent vectors form a basis.