Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Operations with Linear Applications

Description

This lecture covers operations with linear applications in the context of vector spaces. It explains how to define the sum, composition, and other operations between linear applications, providing examples to illustrate the concepts.

In MOOC

Linear Algebra (Part 2)

Official source

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 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 .

Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.

Linear span

In mathematics, the linear span (also called the linear hull or just span) of a set S of vectors (from a vector space), denoted span(S), is defined as the set of all linear combinations of the vectors in S. For example, two linearly independent vectors span a plane. The linear span can be characterized either as the intersection of all linear subspaces that contain S, or as the smallest subspace containing S. The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules.

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.

Related lectures (41)

Linear Algebra: Matrices and Linear Applications

Covers matrices, linear applications, vector spaces, and bijective functions.

Vector Calculus in 3D

Covers the concept of 3D vector space, scalar product, bases, orthogonality, and projections.

Linear Applications: Kernel MOOC: Linear Algebra (Part 2)

Introduces the kernel of a linear application and its properties.

Orthogonal Projection: Vector Decomposition

Explains orthogonal projection and vector decomposition with examples in particle trajectory analysis.

Hermitian Forms: Definition and Properties