vector space

A vector space is a collection of objects called vectors, together with two operations: adding vectors to each other and multiplying vectors by scalars (numbers) from some underlying field, such as the real or complex numbers. These operations must satisfy the usual linearity rules.

In a vector space, vector addition and scalar multiplication:

• Are always defined within the set.
• Are associative.
• Make vector addition commutative.
• Include a special zero vector that acts as an additive identity.
• Include additive inverses.
• Distribute correctly over both vector addition and scalar addition.
• Are compatible with multiplication in the underlying field.
• Leave vectors unchanged when multiplied by the multiplicative identity of the field.

Core ideas built on top of this definition include subspaces, linear independence, span, bases, and dimension. Typical examples of vector spaces are ordinary n-dimensional real or complex spaces and various function spaces, such as spaces of polynomials or spaces of continuous functions.

These structures form the foundation of linear algebra and are central to scientific computing, optimization methods, and many algorithms in modern machine learning.