A vector space is a collection of vectors that can be added together and multiplied by scalars, subject to certain mathematical rules called axioms. In a vector space, vector addition and scalar multiplication always produce another vector within the same space. Vector spaces form the foundation of linear algebra and are widely used in mathematics, physics, engineering, and computer science.
Examples of vectors in daily life include:
- Position: Used to describe location relative to a reference point.
- Movement: Vectors help us to understand movement in the real world.
- Forces: Show both magnitude and direction in physics.
- Graphics and Animation: Defines shapes and motion in computer graphics.
Foundations
Build a strong understanding of vectors, vector spaces, subspaces, span, and linear independence in linear algebra.
- Linear Algebra
- Vectors in Math
- Properties of Vectors
- Introduction to Vector Spaces
- Vector Algebra
- Vector Subspaces
- Span in Linear Algebra
- Linear Independence of Vectors
Basis, Dimension, Rank & Nullity
Learn how basis, dimension, rank, and nullity describe the structure and properties of vector spaces and matrices.
- Basis and Dimension in Vector Space
- Basis Vectors in Linear Algebra
- Rank of a Matrix
- Rank and Nullity Theorem
- How to Find Column Space of a Matrix
- Transformation Matrix
Inner Product, Orthogonality & Gram Schmidt
Understand inner products, orthogonality, and the Gram Schmidt process for constructing orthonormal bases.