vector space
Sign in to savethe basic algebraic structure of linear algebra; a module over a field, such that its elements can be added together or scaled by elements of the field
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A vector space is a mathematical structure where you can add elements together or scale them by numbers, and these operations follow consistent rules. It's fundamental to linear algebra and appears throughout mathematics, physics, and engineering because many real-world problems—from computer graphics to machine learning—can be understood through this framework.
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Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.
In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field.