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of matrix transformations on R^2 and R^3\ 11:09 Symmetry about the line y = kx, Problem 2 27:06 Rotation by 90 degrees about the origin 38:08 Rotation by the angle α about the origin 47:42 Expansion, compression, scaling, and shear 59:25 Plane symmetry in the 3-space, Problem 3 1:12:02 Projections on planes in the 3-space, Problem 4 1:19:55 Symmetry about a given plane, Problem 5 1:38:47 Projection on a given plane, Problem 6 2:01:56 Rotations in the 3-space, Problem 7 of matrix transformations\ 2:25:23 What kind of properties we will discuss 2:30:38 What happens with vector subspaces and affine subspaces under linear transfo 2:38:51 Parallel lines transform into parallel lines, Problem 1 2:49:54 Transformations of straight lines, Problem 2 3:02:39 Change of area (volume) under linear operators in the plane (space) 3:22:51 Change of area under linear transformations, Problem 3 3:35:44 Compositions of linear transformations 3:49:49 How to obtain the standard matrix of a composition of linear transformations 4:00:59 Why does it work_ 4:17:04 Compositions of linear transformations, Problem 4 4:33:56 Compositions of linear transformations, Problem 5 linear transformations in different bases\ 4:59:43 Linear transformations between two linear spaces 5:06:41 Linear transformations, Problem 1 5:38:06 Linear transformations, Problem 2 6:08:14 Linear transformations, Problem 3 6:42:54 Linear transformations, Problem 4 7:26:02 Linear transformations, Problem 5 7:45:07 Linear transformations in different bases, Problem 6 8:03:52 Linear transformations in different bases 8:12:10 Linear transformations in different bases, Problem 7 8:36:54 Linear transformations in different bases, Problem 8 8:53:38 Linear transformations in different bases, Problem 9 9:08:42 Linear transformations, Problem 10 9:16:18 Linear transformations, Problem 11 –Schmidt process\ 9:32:02 Dot product and orthogonality until now 9:44:54 Orthonormal bases are awesome 9:52:59 Orthonormal bases are awesome, Reason 1_ distance 9:55:26 Orthonormal bases are awesome, Reason 2_ dot product 9:59:57 Orthonormal bases are awesome, Reason 3_ transition matrix 10:03:44 Orthonormal bases are awesome, Reason 4_ coordinates 10:08:27 Coordinates in ON bases, Problem 1 10:35:47 Coordinates in orthogonal bases, Theorem and proof 10:43:55 Each orthogonal set is linearly independent, Proof 10:49:58 Coordinates in orthogonal bases, Problem 2 11:10:49 Orthonormal bases, Problem 3 11:17:49 Projection Theorem 1 11:31:42 Projection Theorem 2 11:54:46 Projection Formula, an illustration in the 3-space
