0:00 Relations
2:28 Reflexive, symmetric, and transitive relations
10:01 Equivalence relations. Examples
16:17 Equivalence classes
19:31 Partitioning a set into equivalence classes
29:44 Equivalence relation on a vector space
40:14 Examples of equivalence classes (cosets) v+U
43:35 Quotient space V/U
55:43 Canonical Projection V→V/U
59:34 dim(V/U) = dim(V) - dim(U)
1:01:13 Basis of the quotient space
1:22:28 Relationship between quotient spaces and direct sums
We begin this lecture by presenting the machinery of equivalence relations. Then we use this tool to construct the quotient V/U of a vector space V by its subspace U. If we think that a direct sum of vector spaces is analogous to addition, then taking quotients is analogous to subtraction for vector spaces.
We show how to construct a basis of the quotient space V/U.
This is a lecture in the "Linear Algebra" course for students specializing in mathematics.
