### 2240 class highlights

Fri 5/3 Final Research Sessions 2-4:30
Wed 5/1 Work on the final project and/or exam corrections. I'll be in class to help!
Mon 4/29 exam 2 corrections,
final project slides, final project criteria, final project rubric, library sources: When Life Is Linear : From Computer Graphics to Bracketology, Pandora, Avengers: Engame
evaluations
• Wed 4/24 Exam 2
• Wed 4/17 Take out the final project handout and the eigenvector worksheet.
Applications to mathematical physics, quantum chemistry..., Standing wave, Eigenfunction, Tacoma Narrows
Google Scholar: eigenvalue in mathematics education research
THE \$25,000,000,000 EIGENVECTOR by Kurt Bryan and Tanya Leise
final research presentations
Hamburger earmuffs and the pickle matrix
full guidelines, topics and sample projects introduction to LaTex.
review activity
Test 2 review, topics to study

• Mon 4/15
Clicker in Chapter 5 #12-
diagonalization, diagonalization and computer graphics and eigenfaces
definitions matching
• Wed 4/9
Clicker in Chapter 5 #4-10 Highlight predator prey, predator predator or cooperative systems (where cooperation leads to sustainability)
Geometry of Eigenvectors
Ex1:=Matrix([[0,1],[1,0]]);
Eigenvalues(Ex1);
Eigenvectors(Ex1);
Ex2:=Matrix([[0,1],[-1,0]]);
Ex3:=Matrix([[-1,0],[0,-1]]);
Ex4:=Matrix([[1/2,1/2],[1/2,1/2]]);

Horizontal shear Matrix([[1,k],[0,1]])
Continue Dynamical Systems and Eigenvectors
comic http://brownsharpie.courtneygibbons.org/comic/guest-artist-little-pete-is-emo/
eigensheep comic
Clicker in Chapter 5 #11

• Mon 4/8
Review algebra of eigenvalues and eigenvectors. worksheet on eigenvector decomposition
M := Matrix([[6/10,4/10],[-125/1000,12/10]]);
Eigenvectors(M);

Eigenvector decomposition
Application: Foxes and Rabbits
Clicker in Chapter 5 #1-3
Compare with Dynamical Systems and Eigenvectors first example. Never cross me!
• Wed 4/3
nullspace null=me
clicker in 2.8 and real-life applications.
Write down an example of a 2x2 matrix representation of each of the following transformations: reflection, dilation, and shear
eigenvalues and eigenvectors. Apply the characteristic equation to your examples. Eigenvalues of triangular matrices are on the diagonal.
Matrix([[2,1],[1,2]])
M := Matrix([[2,1],[1,2]]); Eigenvectors(M);
Introduce foxes and rabbits: M := Matrix([[6/10,4/10],[-125/1000,12/10]]);
Eigenvectors(M);

• Mon 4/1
Clicker questions in chapter 3 #10
If space is the final frontier, then what's a subspace [Paramount and CBS]?
basis, null space and column space, 2.8 worksheet
• Wed Mar 27
Review Laplace expansion
Clicker questions in chapter 3 #4-6
Mention google searches: application of determinants in various fields including eight queens in game theory and volumetric strain in geology
The relationship of row operations to the geometry of determinants
Clicker questions in chapter 3 #7-9
final project products, peer review and rubric

• Mon Mar 25 Review Maple commands on determinants. worksheet
LaTex Beamer slides
determinator comic
matrix operations and determinants
• Wed Mar 20
worksheet on definitions, Review and continue yoda2.mw. last glossary.
Clicker questions in 2.7 continued
begin Clicker questions in chapter 3 #1-3
2x2 and 3x3 diagonals methods and Laplace's expansion (1772 - expanding on Vandermonde's method) method in general. [general history dates to the Chinese and Leibniz]
M:=Matrix([[a,b,c],[d,e,f],[g,h,i]]);
Determinant(M); MatrixInverse(M);
M:=Matrix([[a,b,c,d],[e,f,g,h],[i,j,k,l],[m,n,o,p]]);
Determinant(M); MatrixInverse(M);

• Mon Mar 18
Review geometric transformations of the plane
Computer graphics demo [2.7] Examples 1 and 2
Clicker questions in 2.7 #1-2
rotation matrix and 6.1
Application of 2.7 and 6.1: Keeping a car on a racetrack
Computer graphics demo [2.7] Examples 3-5
Begin Yoda (via the file yoda2.mw) with data from Kecskemeti B. Zoltan (Lucasfilm LTD) as on Tim's page
exam corrections
• Wed Mar 13 Exam 1

• Mon Mar 11
Review and introduction to linear transformations
geometric transformations of the plane
We can also apply the product matrix M repeatedly to create a simple animation: Here is a movie created by 20 frames of applying URT(triangle). Careful of the order! Careful of the order - if I do T.R.U(triangle) instead.
Clicker question in 1.3, 1.4, 1.5, 1.7 #11-
Bring up solutions to Problem Set 3 on ASULearn and compare with your problem set
worksheet on definitions
review slides, sample partial exam, study guide for exam 1
• Wed Feb 27
Clicker in 2.1-2.3 #20-22
Not on exam 1: Glossary of terms: Review and introduction to linear transformations, Guess the transformation, worksheet VLA Package from Visual Linear Algebra by Herman and Pepe. In the process, discuss that the first column of the matrix representation is the same as the output of the unit x vector.
Sheared Sheap comic from our book, general geometric transformations on R2 [1.8, 1.9, 2.7], review the unit circle

• Mon Feb 25
2.2 #21, review 2.3 and chapter 1 definitions, and theorem 8
Clicker in 2.1-2.3 #11-13
Review and continue condition number and Hill cipher
Continue Maple file on condition number and Hill cipher, PDF
Clicker in 2.1-2.3 #14-19
• Wed Feb 20
Review 2.1 #23 (multiplicative argument and then pivot argument). Example where A doesn't have to be square---3x2 matrix A.
Clicker in 2.1-2.3 #8-10
Comic: associativity superpowers
Assume that A (square) has an inverse. What else can you say?
Theorem 8 in 2.3 [without linear transformations], not invertible but still square, not square? List relevant examples and course overview
What makes a matrix invertible
Begin applications of 2.1-2.3 via course overview and condition number
Maple file on condition number and Hill cipher, PDF
Begin the Hill cipher history

• Mon Feb 18
2.2 Algebra: Inverse of a matrix
Finish Clicker in 2.1-2.3 #5
twobytwo := Matrix([[a, b], [c, d]]);
MatrixInverse(twobytwo);
MatrixInverse(twobytwo).twobytwo
simplify(%)

1.3 and 1.7 vector and matrix equations, ASULearn solutions and optional
Applications of multiplication and the inverse (if it exists). The first of these should look familiar!
success
Clicker in 2.1-2.3 #6-7
• Wed Feb 13
Review span and l.i. and mention video and solutions on ASULearn
glossary for 2.1-2.3, glossary for 1.3, 1.4, 1.5, 1.7, glossary for 1.1 and 1.2
Begin Chapter 2:
Then Clicker in 2.1-2.3 #1-4
matrix multiplication and matrix algebra. AB not BA...
Introduce transpose of a matrix via Wikipedia, including Arthur Cayley. Applications including least squares estimates, such as in linear regression, data given as rows (like Yoda
Begin Clicker in 2.1-2.3 #5

• Mon Feb 11
1.3 and 1.7 vector and matrix equations
Review theorem 4 in 1.4. theorem in 1.7
Clicker question in 1.3, 1.4, 1.5, 1.7 #8-10
dependence comic. Shin Takahashi and Iroha Inoue The Manga Guide to Linear Algebra
Roll, Yaw, Pitch by ZeroOne cc 3.0, and Gimbal lock on Apollo missions
Continue Maple file on span and l.i.
Review 1.1, 1.2, 1.3, 1.4, 1.5, 1.7
Continue span and l.i.
• Wed Feb 6
Clicker in 1.3-1.7 # 6, 20
#1 on span and l.i.
Clicker in 1.3-1.7 #7 to motivate 1.7 and redundancy
1.3 and 1.7 vector and matrix equations
Review theorem 4 in 1.4. theorem in 1.7
Continue with span and l.i. in R^2 #2-4 and discuss 5.
Begin Maple file on span and l.i.

• Mon Feb 4
Discuss REUs, majoring or minoring in mathematics and fall schedule. Review Clicker in 1.3-1.7 #4
Decimals (don't use in Maple) and fractions. Geometry of the columns as a plane in R^4, of the rows as 4 lines in R^2 intersecting in the point (40,60).
Second glossary. Review 1.3 and 1.4
Completion in Moodle, Grades in ASULearn, Solutions
Theorem 4 in 1.4
Clicker in 1.3-1.7 #5 and in Maple, including equation for the span, its geometry, and finding vectors inside and outside of it by using and modifying the diagonal of the parallelogram.
1.5: vector parametrization equations of homogeneous and non-homogeneous equations.
parallelvectorline movie. Introduce t*vector1 + vector2 is the collection of vectors that end on the line parallel to vector 1 and through the tip of vector 2.
• Wed Jan 29 Collect hw and take out glossaries and i-clicker questions. ASULearn solutions and discuss problem sets George Polya: How to Solve it
spand3dmovie
Clicker in 1.3-1.7 #1-3
Peer review hw
Begin 1.4.
Clicker in 1.3-1.7 #4
with(LinearAlgebra): with(plots):
Coff:=Matrix([[.3,.4,36],[.2,.3,26],[.2,.2,20],[.3,.1,18]]);
ReducedRowEchelonForm(Coff);
Coffraction:=Matrix([[3/10,4/10,36],[2/10,3/10,26],[2/10,2/10,20],[3/10,1/10,18]]);
ReducedRowEchelonForm(Coffraction);

Decimals (don't use in Maple) and fractions.
Theorem 4 in 1.4

• Mon Jan 27
3 unknowns in R^3, advice from previous students.
1.3 and Glossary for rest of chapter 1
Maple for span
Foxtrot vector addition comic by Bill Amend. November 14, 1999.
• Wed Jan 23 Collect hw and take out the glossary of terms, i-clicker questions, and day 1 handout with the problem set guidelines.
Clicker in 1.1 and 1.2 #2 onwards
Problem set 1

• Wed Jan 16
Turn in the hw. Review EBG #5. Prove using geometry of lines that the number of solutions of a system with 2 equations and 2 unknowns is 0, 1 or infinite.
Engagement with the the i-clickers Hand out i-clicker questions and do Clicker in 1.1 and 1.2 #1.
Gaussian and Gauss-Jordan or reduced row echelon form in general.
Review Gaussian and Gauss-Jordan for 3 equations and 2 unknowns in R2. section 1.2, focusing on algebraic and geometric perspectives and solving using by-hand elimination of systems of equations with 3 unknowns. Follow up with Maple commands and visualization: ReducedRowEchelon and GaussianElimination as well as implicitplot3d in Maple (like on the handout):
Drawing the line comic.
Parametrize x+y+z=1. Maple
with(LinearAlgebra): with(plots): Ex1:=Matrix([[1,-2,1,2],[1,1,-2,3],[-2,1,1,1]]);
implicitplot3d({x-2*y+z=2, x+y-2*z=3, (-2)*x+y+z=1}, x = -4 .. 4, y = -4 .. 4, z = -4 .. 4);
Ex2:=Matrix([[1,2,3,3],[2,-1,-4,1],[1,1,-1,0]]);
implicitplot3d({x+2*y+3*z=3,2*x-y-4*z=1,x+y-z=0}, x=-4..4,y=-4..4,z=-4..4);
Ex3:=Matrix([[1,2,3,0],[1,2,4,4],[2,4,7,4]]);
implicitplot3d({x+2*y+3*z = 0, x+2*y+4*z = 4, 2*x+4*y+7*z = 4}, x = -13 .. -5, y = -1/4 .. 1/4, z = 3 .. 5, color = yellow);
Ex4:=Matrix([[1,3,4,k],[2,8,9,0],[10,10,10,5],[5,5,5,5]]);
GaussianElimination(Ex4);
Ex4a:=Matrix([[1,3,4,k],[2,8,9,0],[10,10,10,5]]);
GaussianElimination(Ex4a);

Highlight equations with 3 unknowns with infinite solutions, one solution and no solutions in R3, and the corresponding geometry, as we review new terminology and glossary of terms

• Mon Jan 14 UTAustinXLinearAlgebra.mov
Course intro slides # 1 and 2
Work on the introduction to linear algebra handout motivated from Evelyn Boyd Granville's favorite problem (#1-3). At the same time, begin 1.1 (and some of the words in 1.2) including geometric perspectives, by-hand algebraic EBG#3, Gaussian Elimination and EBG #5 and pivots, solutions, plotting and geometry, parametrization and GaussianElimination in Maple for systems with 2 unknowns in R2.
Evelyn Boyd Granville #3:
with(LinearAlgebra): with(plots):
implicitplot({x+y=17, 4*x+2*y=48},x=-10..10, y = 0..40);
EBG3:=Matrix([[1,1,17],[4,2,48]]);
GaussianElimination(EBG3);
ReducedRowEchelonForm(EBG3);

Evelyn Boyd Granville #4: using the slope of the lines, versus full pivots in Gaussian (r2'=-4 r1 + r2):
EBG4:=Matrix([[1,1,a],[4,2,b]]);
GaussianElimination(EBG4);

Course intro slides continued.
How to get to the main calendar page: google Dr. Sarah / click on webpage / then 2240. Discuss webpages, homework and Polya's How to Solve it
Vocabulary/terms/ASULearn glossary

Evelyn Boyd Granville #5 with k as an unknown but constant coefficient.
EBG#3, Gaussian Elimination and EBG #5
EBG5:=Matrix([[1,k,0],[k,1,0]]);
GaussianElimination(EBG5);
ReducedRowEchelonForm(EBG5);