Coordinate Vectors, Coordinate Mappings, Change of Coordinates Matrix, Dimension of a Vector Space

(a.k.a. Differential Equations with Linear Algebra, Lecture 19B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 19B).

#vectorspace #basis #coordinatevectors

(0:00) Introduction
(0:40) Examples and Nonexamples of Bases of 3-space
(6:20) Unique Representation Theorem (to guarantee that coordinate vectors are well-defined)
(8:03) Coordinate vector with respect to an ordered basis of a vector space V
(10:15) Coordinate mapping from V to R^n is a linear transformation (assuming V has a basis with n vectors). It is operation-preserving.
(14:05) The coordinate mapping is an isomorphism. We say the vector space V is isomorphic to R^n.
(15:54) Example: find a coordinate vector of a vector in R^2 with respect to an ordered basis. Solve with an inverse matrix. Use the shortcut formula for the inverse of a 2 x 2 invertible matrix (assuming its determinant is nonzero).
(21:14) Three Pictures to Visualize This
(26:03) Change of Coordinates Matrix in Euclidean Space
(30:34) Definition of dimension of a vector space
(34:01) Example of an infinite dimensional vector space
(36:06) Basis Theorem: If dim(V) = n (positive), (i) then any linearly independent set with n elements is a basis of V and (ii) any spanning set with n elements is a basis of V.

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[georgesadler]

Professor Kinney, thank you for another theoretical video/lecture on Coordinate Vectors/Mapping, Change of Coordinate Matrix and the Dimension of Vector Space. All these topics are tied to abstract Linear Algebra. The main idea of this lecture is for students in Mathematics/Science to think abstractly and be able to write proofs.