LinA: Linear Algebra [curr]

Computer&Projector	CP-Probs	Time	Memory/Telepathy	Blackboard
Dawson/Jake	Allan	Pietro/Katie	Katie	Xavier/everybody

Optional: LinA IOP (Individual Optional Project)

...is due, slid u n d e r room 402 Little Hall my office door (Little Hall 402, Northeast corner) no later than [2PM, Thursday, 07Dec2023].

This IOP must be carefully typed, but diagrams may be hand-drawn.

At all times have a paper copy you can hand-in; I do NOT accept electronic versions. Print out a copy each day, so that you always have the latest version to hand-in; this, in case your printer or computer fails. (You are too old for My dog ate my homework.)

Please follow the guidelines on the Checklist (pdf, 3pages) to earn full credit.

Inner product spaces

In one convenient spot: All 22 LinA quizzes (pdf) [Tuesday, 05Dec2023]
Also: Abbreviations-to-memorize.
Our Class-T was preceded by practice Class-T.
w:IP spaces and w:Gram-Schmidt algorithm. An example of Gram-Schmidtification (txt).
Examples of Gram-Schmidt in R³ and Gram-Schmidt in R⁴.
And Gram-Schmidt in R⁴.
Pamphlet on complex inner-products (pdf).
A template for computing a 2x2 Positive-definite matrix for an inner-product..
How much would you pay for a 3x3 positive-definite example?
I'll bet we can Ortho-project on a plane!
Least Squares and matrices (pdf): Derivation of a formula for Least-squares fitting of a line to data points (also has HW problems). Available are two examples of a more-general curve-fitting formula (txt).
Pamphlet Differentiating a bilinear function (pdf) shows how to generalize the Product Rule to other bilinear functions.
Optional: Ring Basics.

Vector spaces

Autumn2023: LinA syllabus
Three matrices: Same Char-Poly but different Min-Polys.
Current:
Have read the beginning of w:Characteristic polynomial. The defn we'll use is in the Cayley-Hamilton pamphlet (pdf).
An example of computing an eigenbasis.
If time permits, we will discuss Jordan Canonical Form (JCF), In particular, I would like you to know the stmt and proof of the Block Upper-Triangular--matrix Lemma [just page 4 in JCF].
Prof. K's LinAlg-Notes (pdf)
Linear Recurrence using matrices (pdf) solves a fibonacci-like recurrence; it makes a cryptic reference to Jordan Canonical Form. Its method is to attempt to diagonalize a matrix.

Examples: a 2x2 diagonalizable matrix. And a 3x3 matrix with only 2 dimenions of eigenvalues. Also powers of a diagonalizable 4x4 matrix.

Finally, here are examples showing Trace and char-poly are preserved under conjugation.
Our wholly expialadocious Class-S is available for cogitation and edification.
An overview of the Sign of a permutation.
Available are more extensive notes on permutations,
Here we explore N×N determinants (txt) and an application of Cramer's “Rule”, i.e Theorem (txt).
Abstract properties of determinants.

Work through w:Determinant.

Please read the first few paragraphs of: w:Bilinear_map and w:Bilinear_form, as well as w:Multilinear_map and w:Alternating_multilinear_map.
A proof of the Rank-Nullity Theorem (pdf), and also a proof that RREF is unique (pdf), stated appropriately.
Voila! Examples of RREF over Rationals and RREF over Zed_p (plus Change-of-Basis) and RREF over Complex numbers.
Example: Computing a matrix-inverse
Examples on obtaining a basis for column-span, and Change-of-Basis (Maple).
Computing bases for the LeftNullspace, ColumnSpan and RowSpan of a 4x7 matrix. Also available is a Gauss-Jordan calculator.
Gaussian elimination Animation
Future: Rotation-matrix cartoon: w:The Circle group and w:Rotations in 3D space. Do we like w:Rotations in N-dim'al space?
The delightful Class-R is online for your soln-posting pleasure.
The Euclidean algorithm can be presented in table-form; I call this form the Lightning-bolt algorithm (pdf), because the update-rule looks like a lightning-bolt (used thrice). Here is a practice sheet for LBolt (pdf).
What is Gambling doing here?. Aha! now we know...
BingChat —a Cautionary Tale...
Past: the Math-Greek alphabet (pdf).

This demo page illustrates these linear transformations: Dilations, Rotations and Shears. It also shows the affine transformation of Translation.

Please peruse the Wikipedia pages that define Group, Ring, Field as well as Set builder notation.

General Info

Our LinA class has a LISTSERV archive. I will email to each student how to post-to and read-from the Archive. (The archive is at a private URL, only for the use of the folks in our class.)

Quantifiers ∀ and ∃ (“for all” and “there exists”) are like nitroglycerin, in that one little mis-step leads to the whole thing blowing up in your face.

There is no partial credit when it comes to Explosives and Quantifiers.

-JLF King

Michael's Matrices

(defun Num-GL-matrices (P N &aux VS-Card NumMatrices NumInvertible Prob) "Our vectorspace, VS, is N-dim'al over Zed_P, where P is prime. Our NxN matrices are points in VS of dimension P²." (setq VS-Card (expt P N) NumMatrices (expt VS-Card N) ; # of ordered lists of N vectors; a matrix. NumInvertible (iter (for kMO :below N) (for p^kMO :first 1 :then p^k) (for p^k = (* p^kMO p)) (multiply (- VS-Card p^kMO)) ) ) (setq Prob (/ NumInvertible NumMatrices)) (when (> *JK-PRINT* 5) (format t "~%Prob(GL_~D(Zed_~D)) = ~A ≈≈ ~F.~%" N P Prob Prob) (format t "~%Prob-of-singular = ~A ≈≈ ~:*~F.~%" (- 1 Prob)) ) (list NumInvertible Prob) ;; Return-value )

% (Num-GL-matrices 17 3) Prob(GL_3(Zed_17)) = 22634496/24137569 ≈≈ 0.9377289. Prob-of-singular = 1503073/24137569 ≈≈ 0.0622711. (111203278848 22634496/24137569) % (iter (for N :from 1 :to 12) (format t "~%~2D: ~101D" N (first (Num-GL-matrices 5 N)))) 1: 4 2: 480 3: 1488000 4: 116064000000 5: 226614960000000000 6: 11064475422000000000000000 7: 13506266841692625000000000000000000 8: 412177498341354683437500000000000000000000000 9: 314466314168148447161790527343750000000000000000000000000 10: 5997968329750020529620924720314025878906250000000000000000000000000000 11: 2860054114776434486551525246466392329633235931396484375000000000000000000000000000000 12: 34094501770277653127041454190627871991478120675310492515563964843750000000000000000000000000000000000

LinA Assignment for Add/Drop week

In order to determine what you need to review over the weekend, please use 80 minutes do to as many problems as you can from this self-test of prerequisite knowledge (pdf). Then use the internet and your textbooks to grade yourself. This will help you determine what you wish to review.
Memorize the Math-Greek alphabet (pdf).

Author:	Friedberg, Insel, Spence	ISBN:	978-0134860244
Year:	2019	Publisher:	Pearson

(Abstract) Linear Algebra