Lecture 22. Composition of Linear Transformations

0:00 Composition of functions 6:52 Composition of two linear transformations is linear 12:05 Matrix of a composition of two transformations 21:01 Example: computing a product of two matrices 26:25 General formula for the product of two matrices 28:51 Compatibility condition on the sizes of matrix factors 31:35 Example: matrix of a composition of a rotation with a reflection 38:20 Algebraic properties of matrix product 42:53 Identity matrix 46:55 Matrix product is associative, but not commutative 54:16 Formula for the transpose of a product 1:00:42 Product of rotation matrices and trigonometric identities In this lecture, we explore the connection between compositions of linear transformations and multiplication of matrices. As an application of linear algebra, we derive trigonometric identities. This is a lecture in the "Linear Algebra" course for students specializing in mathematics.