Abstract Algebra, Lec 33A: Review Linear Algebra: Linear Independence, Basis, Dimension, Examples

Abstract Algebra, Lecture 33A. Field Theory and Galois Theory, Part 2.

(0:00) Fun little sports stories to come in future lectures.
(0:16) Lecture plan.
(0:55) Review the definition of a vector space V over a field F.
(5:06) Linear dependence and linear independence.
(7:43) Examples in R^3 using invertible matrix theorem (thought of in terms of determinants and row operations to reduced row echelon form).
(14:34) Definition of a basis of a vector space.
(16:00) If two finite sets are bases for the same vector space V over the same field F, then the sets have the same number of elements (same cardinality). This justifies the definition of the dimension of a vector space.
(17:07) The choice of basis helps you solve various problems in the applications of linear algebra.
(18:39) The solution set of a second-order linear differential equation is a two-dimensional vector space over R.
(26:30) Other worthwhile facts to know (bases are maximal linearly independent spanning sets and minimal spanning sets that are linearly independent).

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