Vector spaces | Mathematical structures | Vectors (mathematics and physics) | Group theory

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. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations. Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension. Many vector spaces that are considered in mathematics are also endowed with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras and Lie algebras. This is also the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. (Wikipedia).

This video explains the definition of a vector space and provides examples of vector spaces.

From playlist Vector Spaces

What is a Vector Space? Definition of a Vector space.

From playlist Linear Algebra

What is a Vector Space? (Abstract Algebra)

Vector spaces are one of the fundamental objects you study in abstract algebra. They are a significant generalization of the 2- and 3-dimensional vectors you study in science. In this lesson we talk about the definition of a vector space and give a few surprising examples. Be sure to su

From playlist Abstract Algebra

linear algebra vector space (25 examples)

Vector Spaces. Definition and 25 examples. Featuring Span and Nul. Hopefully after this video vector spaces won't seem so mysterious any more! Check out my Vector Space playlist: https://www.youtube.com/watch?v=mU7DHh6KNzI&list=PLJb1qAQIrmmClZt_Jr192Dc_5I2J3vtYB Subscribe to my channel:

From playlist Vector Spaces

The formal definition of a vector space.

From playlist Linear Algebra Done Right

Multivariable Calculus | The notion of a vector and its length.

We define the notion of a vector as it relates to multivariable calculus and define its length. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Vectors for Multivariable Calculus

Vector spaces | Lecture 16 | Matrix Algebra for Engineers

Definition of a vector space. Join me on Coursera: https://www.coursera.org/learn/matrix-algebra-engineers Lecture notes at http://www.math.ust.hk/~machas/matrix-algebra-for-engineers.pdf Subscribe to my channel: http://www.youtube.com/user/jchasnov?sub_confirmation=1

From playlist Matrix Algebra for Engineers

After our introduction to matrices and vectors and our first deeper dive into matrices, it is time for us to start the deeper dive into vectors. Vector spaces can be vectors, matrices, and even function. In this video I talk about vector spaces, subspaces, and the porperties of vector sp

From playlist Introducing linear algebra

Now we know about vector spaces, so it's time to learn how to form something called a basis for that vector space. This is a set of linearly independent vectors that can be used as building blocks to make any other vector in the space. Let's take a closer look at this, as well as the dimen

From playlist Mathematics (All Of It)

What Is A Tensor Lesson #1: Elementary vector spaces

We define a vector space and lay the foundation of a solid understanding of tensors.

From playlist What is a Tensor?

A Swift Introduction to Spacetime Algebra

This video is a fast-paced introduction to Spacetime Algebra (STA), which is the geometric algebra of Minkowski space. In it, we figure out what the problems are with the way introductory textbooks usually describe special relativity and how we can solve those problems by using spacetime

From playlist Miscellaneous Math

[Lesson 11] QED Prerequisites - Tensor Product Spaces

We take a detour from the Angular Momentum Mind Map to cover the important topic of Tensor Product spaces in the Dirac Formalism. In quantum mechanics, the notion of tensors is hidden under the hood of the formalism and this lesson opens that hood. The goal is to make us confident that we

From playlist QED- Prerequisite Topics

[Lesson 1] QED Prerequisites Dirac Formalism Part I (redux)

(Editorial repair made in this version) This lecture is the first in a series of topics related to QED prerequisite material. I will be selecting some topics that students are often not clear about when arriving at QED. These topics cover a wide variety of material in elementary quantum m

From playlist QED- Prerequisite Topics

9. Independence, Basis, and Dimension

MIT 18.06 Linear Algebra, Spring 2005 Instructor: Gilbert Strang View the complete course: http://ocw.mit.edu/18-06S05 YouTube Playlist: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8 9. Independence, Basis, and Dimension License: Creative Commons BY-NC-SA More information at h

From playlist MIT 18.06 Linear Algebra, Spring 2005

QED Prerequisites Geometric Algebra: Spacetime.

In this lesson we continue our reading of an excellent paper on Geometric Algebra and spacetime algebra. The paper can be found here: https://arxiv.org/abs/1411.5002 We will cover section 3.1 and begin section 3.2. This material includes our first expansion of the vector space of spacet

From playlist QED- Prerequisite Topics

What is a Tensor 14: Vector and Tensor Fields

What is a Tensor 14: Vector and Tensor Fields

From playlist What is a Tensor?

What is a Tensor 13: Realization of a Vector Space

What is a Tensor 13: Realization of a Vector Space Note: There is an error at 3:26. The equality I write down is only true for orthonormal basis vectors! There will always be a relationship between (e_\mu, e_\nu) and (e^\mu , e^\nu) but it wont always be as simple as I wrote down! For som

From playlist What is a Tensor?

WildLinAlg17: Rank and Nullity of a Linear Transformation

We begin to discuss linear transformations involving higher dimensions (ie more than three). The kernel and the image are important spaces, or properties of vectors, associated to a linear transformation. The corresponding dimensions are the nullity and the rank, and they satisfy a simple

From playlist A first course in Linear Algebra - N J Wildberger

Lecture 2 | Modern Physics: Quantum Mechanics (Stanford)

Lecture 2 of Leonard Susskind's Modern Physics course concentrating on Quantum Mechanics. Recorded January 21, 2008 at Stanford University. This Stanford Continuing Studies course is the second of a six-quarter sequence of classes exploring the essential theoretical foundations of mode

From playlist Quantum Mechanics Prof. Susskind & Feynman