LA05 Matrix Vocabulary & Matrices as Vector Spaces

Basic Matrix Vocabulary is introduced and the set of all n by m #matrices with real entries is given as an example of #vectorspaces Matrix transpose and the trace of a matrix is defined, and is explained why the set of 2x3 matrices with real entries is a six dimensional vector space.#linearalgebra Subscribe @Shahriari for math videos for college students.
00:00 Introduction
00:15 Definition: Matrix, entries, rows and columns
01:12 The Zero matrix
01:22 Square matrices, diagonal of a matrix
01:41 Identity Matrix
02:00 M_{n x m}(R) or R^{n x m} the set of all n x m matrices
02:51 Scalar multiplication for matrices
03:04 Addition of matrices
03:46 Properties of matrix addition and scalar multiplication
05:33 R^{n x m} is a vector space /R
05:47 Basis and dimension for R^{2 x 3}
06:53 F^{n x m} when F is a field is a vector space/F
07:58 Column and row vectors, transpose of a matrix
08:46 Definition: Symmetric Matrix
08:56 Definition: trace of a square matrix

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 series, general vector spaces and linear transformations are emphasized.

In the first few lectures, a number of mathematical structures are studied, and some common features noticed. From these the notion of a vector space is abstracted out. The later lectures will study vector spaces in the abstract.

For an annotated list of available Linear Algebra videos see

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.