udemy-linear-algebra-and-geometry-2-2021-8-0

to the course\ 0:00 Introduction to the course vector spaces and their subspaces\ 14:11 From abstract to concrete 18:16 From concrete to abstract 22:11 Our prototype 34:47 Formal definition of vector spaces Example 1_ Rn 53:35 Vector spaces, Example 2_ m × n matrices with real entries 1:10:33 Vector spaces, Example 3_ real-valued functions on some interval 1:29:00 Vector spaces, Example 4_ complex numbers 1:43:40 Cancellation property 1:50:33 Two properties of vector spaces; Definition of difference 2:05:09 Some properties of vector spaces 2:21:37 What is a subspace 2:41:56 All the subspaces in R2 2:59:34 All the subspaces in R3 3:10:13 Subspaces, Problem 1 3:19:39 Subspaces, Problem 2 3:46:31 Subspaces, Problem 3 4:08:49 Subspaces, Problem 4 combinations and linear independence\ 4:20:32 Our unifying example 4:27:23 Linear combinations in Part 1 4:38:14 Linear combinations, new stuff. Example 1 4:44:52 Linear combinations Example 2 4:50:34 Linear combinations, Problem 1 5:06:14 Linear combinations, Problem 2 5:34:02 What is a span, definition and some examples 5:43:14 Span, Problem 3 5:59:12 Span, Problem 4 6:15:45 Span, Problem 5 6:17:56 What do we mean by trivial_ 6:24:29 Linear independence and linear dependence 6:39:19 Geometry of linear independence and linear dependence 6:54:40 An important remark on linear independence in Rn 7:05:17 Linearly independent generators, Problem 6 7:29:24 Linear independence in the set of matrices, Problem 7 7:45:48 Linear independence in C^0[−∞, ∞], Problem 8 7:56:25 Vandermonde determinant and polynomials 8:15:06 Linear independence in C^∞(R), Problem 9 8:32:47 Wronskian and linear independence in C∞(R) 8:42:21 Linear independence in C^∞(R), Problem 10 8:48:09 Linear independence in C^∞(R), Problem 11 , basis, and dimension\ 8:57:11 What is a basis and dimension_ 9:06:08 Bases in the 3-space, Problem 1 9:46:44 Bases in the plane and in the 3-space 9:59:56 Bases in the 3-space, Problem 2 10:05:49 Bases in the 4-space, Problem 3 10:25:12 Bases in the 4-space, Problem 4 10:53:34 Bases in the space of polynomials, Problem 5 10:58:11 Coordinates with respect to a basis 11:10:39 Coordinates with respect to a basis are unique 11:19:03 Coordinates in our unifying example 11:24:00 Dimension of a subspace, Problem 6 11:35:03 Bases in a space of functions, Problem 7 of basis\ 11:46:00 Coordinates in different bases 11:52:41 It is easy to recalculate from the standard basis 11:59:26 Transition matrix, a derivation