Image & Kernel (Linear Transformation R^5 to R^2), Composite Transformations & Matrix Multiplication

(a.k.a. Differential Equations with Linear Algebra, Lecture 11B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 11B. #linearalgebra).

#linearalgebra #lineartransformation #matrixmultiplication

(0:00) Linear transformations from higher-dimensional space to a lower-dimensional space
(0:58) Example: T:R^5 to R^2
(4:35) Questions about kernel and image of T (is T one-to-one and/or onto)
(6:22) Row operations on an arbitrary augmented matrix
(7:59) Want to get it to reduced row echelon form (RREF)
(10:32) T is onto
(13:29) Solution set in parametric vector form
(17:05) Particular solution Xp and homogeneous solution Xh. The general solution of the nonhomogeneous equation is their sum Xp + Xh
(18:45) The solution set is a 3-dimensional hyperplane inside 5-dimensional space
(21:50) Definition of the Span of a set of vectors
(23:03) Composite linear transformations (composition)
(26:38) The composition is operation preserving (so it is a linear transformation)
(30:05) Matrix multiplication connection (we compose linear transformations by multiplying matrices)
(33:27) Linear combination representations
(36:07) Definition of matrix multiplication AB in terms of its columns
(37:32) Example of matrix multiplication

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[browncow]

I just want to say how useful I am finding these lectures, and how enormously grateful I am that they are online. The textbook is well-written, but I still got rather lost in the weeds during my first read through. Your explanations help a lot. Covid has had the upside for self-learners like myself, that generous and public-spirited lecturers like you have put a lot of material online. It is really fantastic.

[georgesadler]

Professor Kinney, this is a classic topic that covers Introductory to Advanced topics in Linear Algebra. Every time I studied Linear Algebra, I learned something new. Quantum Physics/Mechanics also uses Linear Algebra to study systems/experiments.

[advancedappliedandpuremath]

Hi Sir hope you are having great time. I want to ask that is it necessary to find rref for fundamental subspaces or we can just conclude them from ref.