**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Linear Algebra: Basis and Canonical Basis

Description

This lecture covers the concept of basis in linear algebra, defining a basis as a set of vectors that are linearly independent and span the vector space. It also introduces the canonical basis, which is a specific basis used to represent vectors in a vector space. The instructor explains how to determine if a set of vectors forms a basis and demonstrates the process of finding the canonical basis. Additionally, the lecture discusses the importance of basis in representing vectors efficiently and accurately in various applications.

Official source

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.

In course

MATH-111(a): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

Instructors (2)

Related concepts (79)

Related lectures (8)

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.

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 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.

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 combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

Acyclic Models: Cup Product and CohomologyMATH-506: Topology IV.b - cohomology rings

Covers the cup product on cohomology, acyclic models, and the universal coefficient theorem.

Bar Construction: Homology Groups and Classifying SpaceMATH-506: Topology IV.b - cohomology rings

Covers the bar construction method, homology groups, classifying space, and the Hopf formula.

Algebraic Kunneth TheoremMATH-506: Topology IV.b - cohomology rings

Covers the Algebraic Kunneth Theorem, explaining chain complexes and cohomology computations.

Characteristic Polynomials and Similar MatricesMATH-111(e): Linear Algebra

Explores characteristic polynomials, similarity of matrices, and eigenvalues in linear transformations.

Linear Algebra: Bases and SpacesMATH-111(e): Linear Algebra

Covers linear independence, bases, and spaces in linear algebra, emphasizing kernel and image spaces.