Projection and Rank | Degrees of Freedom Ch. 4

We've defined degrees of freedom as the "rank of the projection matrix in the quadratic form in the definition of a statistic," and after learning about quadratic forms in chapter 3, it's time to cover projection and rank. Projection is a linear transformation that essentially finds the shadow of a vector on a lower dimensional surface. The rank of a matrix tells us the number of dimensions that matrix outputs onto during matrix-vector multiplication. These concepts formalize the basic idea of degrees of freedom as representing dimensions, which we introduced in chapter 1, which helps us lay the groundwork to cover applications like the t-test, ANOVA, and linear regression in future chapters. This is Chapter 4 in a series on Degrees of Freedom, or, The Geometry of Statistics, which is trying to rigorously but intuitively explain what is easily the most confusing concept in statistics, Degrees of Freedom. Check out the other videos here:    • Degrees of Freedom   I've also previously covered some of the material in this video in my series on Projection:    • Projection Matrix Properties - Projection,...   Chapters: 0:00 Introduction 0:35 Recap of previous chapters 1:10 Agenda 1:40 Matrix multiplication as linear transformation 3:48 Projection introduction 5:57 Projection matrices are idempotent 6:50 Projection matrices are symmetric 10:11 Rank of a matrix 14:38 Quadratic forms as squared lengths of projected vectors 16:00 Projections of the data vector 17:27 Matrix representations of these projections 17:58 Proving these matrices are projection matrices 20:08 Projections of the error vector 21:24 Recap 22:14 Ranks of the J/n and I-J/n matrices 23:11 Taking stock 23:53 Preview of the next set of videos and closing credits Further Reading/Viewing: The Essence of Linear Algebra, by 3Blue1Brown:    • Essence of linear algebra   Lecture on independent vectors from Gilbert Strang's Linear Algebra course:    • 9. Independence, Basis, and Dimension   Lecture on Projection from Gilbert Strang's Linear Algebra course:    • 15. Projections onto Subspaces   Saville, David J., and Graham R. Wood. Statistical Methods: A Geometric Primer. New York, NY: Springer New York, 1996. https://doi.org/10.1007/978-1-4612-07.... Wickens, Thomas D. The Geometry of Multivariate Statistics. Hillsdale, N.J: L. Erlbaum Associates, 1995. Saville, David J., and Graham R. Wood. Statistical Methods: The Geometric Approach. Corr. 3rd print. Springer Texts in Statistics. New York: Springer, 1997. Attributions: 'Sunday Smooth' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au Other music is from the YouTube Audio Library, by artists Alex Hamlin, E's Jammy Jams, and Asher Fulero. Made with Manim: https://www.manim.community/. The source code will be posted at the conclusion of the series. Tips are appreciated! Tip me at: https://ko-fi.com/slevey