4. Linear Maps and Their Matrices

Linear maps (aka linear transformations) and the matrices that represent them - explained geometrically and intuitively. Matrix-vector multiplication explained as function evaluation. The website I used in the the first computer animation of a linear map: https://shad.io/MatVis/ This is the fourth in what will eventually be a sequence of supplementary videos for a linear algebra class that I teach based on my book 𝑇ℎ𝑒 𝐷𝑎𝑟𝑘 𝐴𝑟𝑡 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝐴𝑙𝑔𝑒𝑏𝑟𝑎. See my website https://bravernewmath.com for more information on my various books: 𝐹𝑢𝑙𝑙 𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠, 𝑇ℎ𝑒 𝐷𝑎𝑟𝑘 𝐴𝑟𝑡 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝐴𝑙𝑔𝑒𝑏𝑟𝑎, 𝑃𝑟𝑒𝑐𝑎𝑙𝑐𝑢𝑙𝑢𝑠 𝑀𝑎𝑑𝑒 𝐷𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡, 𝐿𝑜𝑏𝑎𝑐ℎ𝑒𝑣𝑠𝑘𝑖 𝐼𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑒𝑑. The first three are available for sale as paperbacks at Amazon, and as pdfs at Lulu. (The Lobachevski book is available at Amazon and the American Mathematical Society) 0:00 Intro 0:21 What's a map? 5:06 What's a LINEAR map? 6:26 From a Basis to Everything 13:00 Linear Maps Preserve the Origin 13:42 Symbolic Definition of Linearity 15:35 Ghosts of Calculus Past 16:40 Computational Example 18:29 Dynamic Images of Linear Maps 21:14 Matrices as Shorthand Notation for Maps 23:22 Mx-Vector Product as Shorthand for Evaluating a Map 25:55 Mx-Vector Product as a Weighted Sum of Columns 31:39 Rotation Matrices 42:19 Linear Maps from R³ to R³. 48:19 Linear Maps from Rⁿ to Rⁿ. 49:37 Linear Maps from Rⁿ to Rᵐ... 55:50 ...and (m x n) Matrices