Скачать или смотреть LA15 What are scalars? Fields as Scalars for Vector Spaces

The scalars for #vectorspaces can not only be real numbers, but they can be the elements of any field. We consider Q (the #rationalnumbers), C (the #complexnumbers), Q[sqrt{2}] (rationals adjoined square root of 2), and integers modulo a prime number, Z/pZ, as possible sets of scalars for a vector space. Modular addition, multiplication, subtraction, and division are explained.#linearalgebra Subscribe ‪@Shahriari‬ for math videos for college students.
00:00 Introduction
00:10 Requirements for scalars in a vector space
00:49 Definition: Field of numbers
01:50 Example: Field of rational numbers
02:59 Example: Field of Complex numbers (   • LA07 Complex Numbers as a Vector Space and...  )
03:56 Non-example: The integers are not a field
04:34 Example: Field of rationals adjoined square root of 2
06:03 Integers modulo 5, Z/5Z (or Z_5)
08:12 Addition and Subtraction mod 5
10:11 Multiplication and Division mod 5
11:09 Definition: Integers modulo n, Z/nZ or Z_n (   • AG06 Integers Modulo n, (Z/nZ, +), as a Gr...  )
12:11 Example: Integers modulo 6 is not a field
13:24 Example: Integers modulo 5 is a field
14:23 Theorem: Z_n is a field if and only if n is a prime (proof in    • AG07 The Group of Units of integers modulo...  )
15:04 Vocabulary: Vector space /F (read over F) means scalars are from the field F
15:27 Example: (Z_7)^3 is a vector space over Z_7

This is a video in a series of lectures on linear algebra. The series is a rigorous treatment meant for students with no prior exposure to linear algebra. In this full course on undergraduate linear algebra, general vector spaces and linear transformations are emphasized.

For an annotated list of available Linear Algebra videos see
https://pomona.box.com/s/oahafa797tj5...
Linear Algebra YouTube Playlist:   • Linear Algebra, a Retro introduction  

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA
Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and five time winner of Pomona College's Wig teaching award.