Definition
A Vector Space is an algebraic structure consisting of a collection of objects (vectors) that can be added together and multiplied (“scaled”) by numbers (scalars), following a specific set of eight axioms. It is the mathematical “playground” where Linearity is defined and explored.
Why It Matters
Vector spaces are the universal containers for high-dimensional data. Whether you are modeling the pixels in an image, the weights of a neural network, or the forces on a bridge, you are operating within a vector space. Understanding its structure allows you to use Linear Algebra to compress data, find patterns, and solve complex systems of equations.
Core Concepts
- Linear Closure: The requirement that adding any two vectors or scaling a vector results in another vector within the same space.
- Basis and Dimension: A basis is a set of linearly independent vectors that “span” the space; the number of vectors in the basis is the space’s dimension.
- Subspaces: A subset of a vector space that is itself a vector space (e.g., a plane passing through the origin in 3D space).
- Linear Independence: A set of vectors is independent if none can be written as a combination of the others.
- Orthogonality: The concept of perpendicularity within a vector space, often defined by an inner product.