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

Lecture

Untitled

Official source

https://mediaspace.epfl.ch/media/0_p2v24228

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

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.

Basis (linear algebra)

In mathematics, a set B of vectors in a vector space V is called a basis (: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called . Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.

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.

Standard basis

In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors Similarly, the standard basis for the three-dimensional space is formed by vectors Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction.

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say is if the dimension of is finite, and if its dimension is infinite.

Related lectures (43)

Linear Independence and Basis

Explains linear independence, basis, and matrix rank with examples and exercises.

Linear Algebra: Vector Spaces

Explores vector spaces, subspaces, bases, and linear combinations in R² and R³, including free and linked families.

Linear Algebra in R³: Vector Spaces

Explores vector spaces in three dimensions, covering linear combinations, subspaces, and properties of families of vectors in R³.

Linear Dependence and Independence: Properties and Criteria MOOC: Linear Algebra (Part 1)

Covers the properties and criteria of linear dependence and independence in a vector space.

Vector Calculus in 3D