Abstract Vector Spaces, Subspaces, Linear Transformations, Kernel, Image, One to One and Onto LTs

(a.k.a. Differential Equations with Linear Algebra, Lecture 16B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 16B).

#linearalgebra #vectorspace #lineartransformation

(0:00) Most of the lecture is about vector spaces
(0:41) Elementary matrix definition
(1:02) Left multiplication by an elementary matrix will perform the corresponding row operation on a given matrix
(1:44) Example with Mathematica help
(7:42) Finish the calculation of the original determinant
(8:59) Definition of a (real) vector space (where the scalars are real numbers)
(10:23) Vector addition properties
(11:21) Define vector subtraction using additive inverses
(12:04) Scalar multiplication properties
(13:09) Examples of vector spaces
(17:40) Basic properties of vector spaces that can be proved (theorems, not axioms)
(19:40) Definition of a subspace of a vector space and the Subspace Test
(21:51) Definition of the span of a finite set of vectors and the fact that it is a subspace
(24:50) Definition of a linear transformation from one vector space V to another W
(27:29) Definition of one-to-one and onto functions
(29:25) Definition of the kernel and image of a linear transformation
(30:52) Examples of linear transformations
(39:38) Theorems about kernels, images, null spaces, column spaces, one-to-one, and onto
(44:07) Comments about invertibility

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[georgesadler]

Professor Kinney thank you for reviewing Vector Spaces, Subspaces, Linear Transformations, Kernel, Image, One to One and Onto Linear Transformations. Although these topics are theoretical, abstract thinking is needed from start to finish. In the early 1990's, at the University of Maryland College Park, I took a senior/400 level Linear Algebra course that was based on writing proofs for all exercises and theorems