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of basis\ 13:02 Previous example with transition matrix 18:02 Our unifying example 27:25 One more simple example and bases 32:23 Two non-standard bases, Method 1 40:31 Two non-standard bases, Method 2 54:17 How to recalculate coordinates between two non-standard bases_ An algorithm 1:07:22 Change of basis, Problem 1 1:47:02 Change of basis, Problem 2 2:00:44 Change of basis, Problem 3 2:24:58 Change of basis, Problem 4 2:33:31 Change of basis, Problem 5 2:45:36 Change to an orthonormal basis in R^2 space, column space, and nullspace of a matrix\ 3:00:15 What you are going to learn in this section 3:06:04 Row space and column space for a matrix 3:15:57 What are the elementary row operations doing to the row spaces_ 3:35:56 What are the elementary row operations doing to the column spaces_ 3:48:13 Column space, Problem 2 4:01:44 Determining a basis for a span, Problem 3 4:16:18 Determining a basis for a span consisting of a subset of given vectors, Prob 4:32:36 Determining a basis for a span consisting of a subset of given vectors, Prob 4:49:47 A tricky one_ Let rows become columns, Problem 6 5:01:14 A basis in the space of polynomials, Problem 7 5:19:13 Nullspace for a matrix 5:32:22 How to find the nullspace, Problem 8 5:40:56 Nullspace, Problem 9 5:58:47 Nullspace, Problem 10 , nullity, and four fundamental matrix spaces\ 6:23:35 Rank of a matrix 6:29:09 Nullity 6:31:38 Relationship between rank and nullity 6:48:09 Relationship between rank and nullity, Problem 1 6:57:16 Relationship between rank and nullity, Problem 2 7:04:58 Relationship between rank and nullity, Problem 3 7:07:46 Orthogonal complements, Problem 4 7:23:05 Four fundamental matrix spaces 7:29:28 The Fundamental Theorem of Linear Algebra and Gilbert Strang transformations from R^n to R^m\ 7:44:22 What do we mean by linear_ 7:51:28 Some terminology 8:04:31 How to think about functions from Rn to Rm_ 8:17:27 When is a function from Rn to Rm linear_ Approach 1 8:26:21 When is a function from Rn to Rm linear_ Approach 2 8:46:48 When is a function from Rn to Rm linear_ Approach 3 8:59:46 Approaches 2 and 3 are equivalent 9:11:28 Matrix transformations, Problem 1 9:18:01 Image, kernel, and inverse operators, Problem 2 9:47:55 Basis for the image, Problem 3 9:56:42 Kernel, Problem 4 10:06:55 Image and kernel, Problem 5 10:18:30 Inverse operators, Problem 6 10:35:38 Linear transformations, Problem 7 10:45:38 Kernel and geometry, Problem 8 10:55:16 Linear transformations, Problem 9 of matrix transformations on R^2 and R^3\ 11:06:25 Our unifying example_ linear transformations and change of basis 11:21:25 An example with nontrivial kernel 11:32:47 Line symmetries in the plane 11:49:18 Projection on a given vector, Problem 1
