LA15 What are scalars Fields as Scalars for Vector Spaces

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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 Spac... ) • 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... ) • 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 Z/nZ; Z/nZ... ) • 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.

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