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Lecture

Vector Spaces: Definitions and Examples

Description

This lecture introduces the concept of vector spaces, defining them as non-empty sets with defined addition and scalar multiplication operations. It explores examples of vector spaces, including R^n and matrices. The lecture also covers subspaces, kernels, and images of linear transformations.

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Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces. If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K.

Topological vector space

In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness.

IPad

The iPad is a brand of iOS and iPadOS-based tablet computers that are developed by Apple Inc, first introduced on January 27, 2010. The iPad range consists of the original iPad lineup and the flagship products iPad Mini, iPad Air, and iPad Pro. The iPhone's iOS operating system (OS) was initially used for the iPad but in September 2019, its OS was switched to a fork of iOS called iPadOS that has better support for the device's hardware and its user interface is customized for the tablets' larger screens.

Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .

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