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

Lecture

Matrix Operations and Vector Spaces

In course

MATH-111(pi): Linear algebra (flipped classroom)

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications. Cette classe est donné sous forme inversée. ...

Description

This lecture covers elementary matrix operations, including swapping columns, multiplying by a scalar, and adding multiples of columns. It also introduces vector spaces, discussing properties like closure under addition and scalar multiplication, as well as the necessary conditions for a set to be a vector space. The lecture further explores vector subspaces, providing examples and discussing their properties. Additionally, it delves into the concept of invertible matrices and the conditions for a matrix to be invertible, along with operations on matrices and their relation to scalar multiplication. The instructor emphasizes the importance of understanding these fundamental concepts for further studies in linear algebra.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Simone Deparis

Donna Testerman

Official source

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.

Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

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.

Graded vector space

In mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers. For "pure" vector spaces, the concept has been introduced in homological algebra, and it is widely used for graded algebras, which are graded vector spaces with additional structures. Let be the set of non-negative integers.

Ordered vector space

In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Given a vector space over the real numbers and a preorder on the set the pair is called a preordered vector space and we say that the preorder is compatible with the vector space structure of and call a vector preorder on if for all and with the following two axioms are satisfied implies implies If is a partial order compatible with the vector space structure of then is called an ordered vector space and is called a vector partial order on The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure.

Related lectures (44)

Introduction to Vector Spaces

Introduces the concept of vector spaces and covers properties of vectorial subspaces.

Vector Spaces: Definition, R2 MOOC: Linear Algebra (Part 1)

Introduces vector spaces with binary addition and scalar multiplication, exploring geometric examples in R2.

Vector Spaces: Operations and Linear Transformations

Explores vector space operations, linear transformations, matrix representation, and linear applications.

Vector Spaces: Properties and Examples

Explores vector spaces, focusing on properties, examples, and subspaces within a practical exercise on polynomials.

Vector Spaces: Properties and Operations Log in to Mediaspace to watch this video