Reflection

final research presentations and last slide

Chinese, German Gauss, French Laplace, German polymath Hermann Grassman (1809-1877) 1844: The Theory of Linear Extension, a New Branch of Mathematics (extensive magnitudes---effectively linear space via linear combinations, independence, span, dimension, projections.)

Diagonalizability, and applications to computer graphics. Applications to mathematical physics and quantum chemistry.

Making a matrix disappear and then reappear 1,2,3, 4,5,6, 7, 8, 9. Look at

h,P:=Eigenvectors(A)

MatrixInverse(P).A.P

which (ta da) has the eigenvalues on the diagonal (when the columns of P form a basis for R

Applications to mathematical physics, quantum chemistry...

ASULearn evaluation

Spend the rest of class time working on test 3 revisions and the project.

Big picture discussion

Review for test 3

Take questions on the study guide or anything else.

definitions and examples activity

Clicker questions--- eigenvector decomposition (5.6) part 2

Clicker questions---review of eigenvectors #1-5

Dynamical Systems and Eigenvectors remaining examples

final research presentations

Hamburger earmuffs and the pickle matrix

sample project,

full guidelines

THE $25,000,000,000 EIGENVECTOR by Kurt Bryan and Tanya Leise: When Google went online in the late 1990's, one thing that set it apart from other search engines was that its search result listings always seemed deliver the "good stuff" up front. With other search engines you often had to wade through screen after screen of links to irrelevant web pages that just happened to match the search text. Part of the magic behind Google is its PageRank algorithm, which quantitatively rates the importance of each page on the web, allowing Google to rank the pages and thereby present to the user the more important (and typically most relevant and helpful) pages first.

About once a month, Google finds an eigenvector of a matrix that represents the connectivity of the web (of size billions-by-billions) for its pagerank algorithm.

http://languagelog.ldc.upenn.edu/nll/?p=3030

Clicker question on interests

Clicker survey questions

Eigenvector comic 1

eigensheep comic Review Eigenvalues and Eigenvectors and the Eigenvector decomposition Also revisit the black hole matrix.

Why we use the eigenvector decomposition versus high powers of A for longterm behavior (reliability)

Compare with Dynamical Systems and Eigenvectors first example

Clicker questions on eigenvector decomposition (5.6) part 1#1-2

Highlight predator prey, predator predator or cooperative systems (where cooperation leads to sustainability)

Eigenvector comic 2

Clicker questions on eigenvector decomposition (5.6) part 1#3-4 [Solutions: 1. a), 2. c), 3. c), 4. b)]

Review reflection across y=x line via pictures. A few inputs. Where is the output? Is the vector an eigenvector?

>Ex1:=Matrix([[0,1],[1,0]]);

>Eigenvalues(Ex1);

>Eigenvectors(Ex1);

Geometry of Eigenvectors examples 1 and 2 and compare with Maple

>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]]) and via det (A-lamda I)=0. Once given lambda, what is the eigenvector?

Clicker questions in Chapter 3 #9

review 2.8 and nullspace

Clicker questions in 2.8

Catalog description:

-Eigenvalues and applications (2.8, 5.1 and 5.6) (after test 3: chap 7 selections)

Begin 5.1: the algebra of eigenvectors and eigenvalues, and connect to geometry and Maple. Eigenvalues and eigenvectors via the algebra as well as the geometry.

Matrix([[2,1],[1,2]])

M := Matrix([[2,1],[1,2]]);

Clicker questions in 5.1#1-3

Begin 5.6: Eigenvector decomposition for a diagonalizable matrix A_nxn [where the eigenvectors form a basis for all of R

M := Matrix([[6/10,4/10],[-125/1000,12/10]]);

Application: Foxes and Rabbits

subspace, basis, null space and column space

2.8 using the matrix 123,456,789 and finding the Nullspace and ColumnSpace (using 2 methods - reducing the spanning equation with a vector of b1...bn, and separately by examining the pivots of the ORIGINAL matrix.) Two other examples. nullspace.

Clicker questions in Chapter 3 #10

Clicker questions in Chapter 3 #8

3.3 p. 180-181:

The relationship of row operations to the geometry of determinants - row operations can be seen as vertical shear matrices when written as elementary matrix form, which preserve area, volume, etc.

Overview of new material for test 2

subspace.

Clicker questions in Chapter 3 #3

Chapter 3 in Maple via MatrixInverse command for 2x2 and 3x3 matrices and then determinant work, including 2x2 and 3x3 diagonals methods, and Laplace's expansion (1772 - expanding on Vandermonde's method) method in general. [general history dates to 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);

LaTex Beamer slides

Review the diagonal determinant methods for the 123,456,789 matrix and introduce the Laplace expansion. Review that for 4x4 matrix in Maple, only Laplace's method will work.

The determinator comic, which has lots of 0s The connection of row operations to determinants The determinant of A transpose and A triangular (such as in Gaussian form). The determinant of A inverse via the determinant of the product of A and A inverse - and via elementary row operations - so det A non-zero can be added into Theorem 8 in Chapter 2: What Makes a Matrix Invertible. Mention google searches: application of determinants in physics application of determinants in economics application of determinants in chemistry application of determinants in computer science Eight queens and determinants application of determinants in geology: volumetric strain

Clicker questions in Chapter 3 4-7

Review linear transformations of the plane, including homogeneous coordinates.

Comic: associativity superpowers

Clicker questions in 2.7 #5-6

Keeping a car on a racetrack

Clicker questions in 2.7 #7

Review linear transformations of 3-space: 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

Clicker questions in 2.7 #8

Clicker questions in Chapter 3 #1 and 2

Chapter 3 in Maple via MatrixInverse command for 2x2 and 3x3 matrices and then determinant work, including 2x2 and 3x3 diagonals methods, and Laplace's expansion (1772 - expanding on Vandermonde's method) method in general. [general history dates to 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);

LaTex Beamer slides

Review What Makes a Matrix Invertible

Go over 2.3 #11c and 12e on solutions.

Clicker questions in 2.7 #1

Applications of 2.1-2.3:

1.8 (p. 62, 65, & 67-68), 1.9 (p. 70-75), and 2.7 Review Guess the transformation. In the process, discuss that the first column of the matrix representation is the same as the output of the unit x vector, and that invertible matrices will take the plane to the plane (the range is onto the plane), while matrices that are not invertible do not span the entire plane, so they smush the plane (pictures in the plane, etc).

general geometric transformations on R

In the process, review the unit circle

Begin Computer graphics demo [2.7]

Clicker questions in 2.7 #2-4

Clicker questions in 2.3 and Hill Cipher #1-3

Review What Makes a Matrix Invertible

Comic: associativity superpowers

Review linear transformations: Ax=b where A is fixed, x are given like in a code or the plane and we see or use the b outputs. 1 unique solution, 0 and infinite solutions, and 0 and 1 solutions.

Linear transformations in the cipher setting and finish 2.3 via the condition number.

Maple file on Hill Cipher and Condition Number and PDF version

Computer graphics and linear transformations (1.8, 1.9, 2.3 and 2.7): Guess the transformation

Mirror mirror comic and Sheared Sheap comic

Review 2.1 #21

multiply comic, identity comic

Clicker questions in 2.2 #1 and 2 In groups of 2-3 people, assume that A (square) has an inverse. What else can you say about the Gauss-Jordan reduction of A, the columns of A, the pivots of A, or systems of equations involving A as the coefficient matrix? Reason using only each other (no books, notes...).

Theorem 8 in 2.3 [without linear transformations]: A matrix has a unique inverse, if it exists. A matrix with an inverse has Ax=b with unique solution x=A^(-1)b, and then the columns span and are l.i...

What makes a matrix invertible

Discuss what it means for a square matrix that violates one of the statements. Discuss what it means for a matrix that is not square (all bets are off) via counterexamples.

-2.1-2.3 Applications: Hill Cipher, Condition Number and Linear Transformations (2.3, 1.8, 1.9 and 2.7)

Applications: Introduction to Linear Maps

The black hole matrix: maps R^2 into the plane but not onto (the range is the 0 vector).

Dilation by 2 matrix

Linear transformations in the cipher setting:

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |

Applications of 2.1-2.3:

Hill Cipher history

Maple file on Hill Cipher and Condition Number and PDF version

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).

2.2: Multiplicative Inverse for 2x2 matrix:

twobytwo := Matrix([[a, b], [c, d]]);

MatrixInverse(twobytwo);

MatrixInverse(twobytwo).twobytwo

simplify(%)

2.2 Algebra: Inverse of a matrix.

Repeated methodology: multiply by the inverse on both sides, reorder by associativity, cancel A by its inverse, then reduce by the identity to simplify:

Applications of multiplication and the inverse (if it exists).

Clicker in 2.1 and 2.2 continued: #7-8.

Test 1 review part 2

Take review questions for test 1. Continue Chapter 2.

Continue via Clicker questions in 2.1 3-4

Image 1 Image 2 Image 3 Image 4 Image 5 Image 6 Image 7.

Continue matrix algebra via Clicker questions in 2.1 5 and 6 (full list: Clicker questions in 2.1)

Matrix multiplication 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).

dependence comic

Clicker review questions

Test 1 review part 1

Begin Chapter 2:

Continue via Clicker questions in 2.1 1-2

Clicker question in 1.3 and 1.5 #5

discuss what happens when we correctly use GaussianElimination(s13n15extension) - write out the equation of the plane that the vectors span.

s13n15extension:=Matrix([[1,-5,b1],[3,-8,b2],[-1,2,b3]]);

GaussianElimination(s13n15extension);

Choose a vector that violates this equation to span all of R^3 instead of the plane and plot:

M:=Matrix([[1,-5,0,b1],[3,-8,0,b2],[-1,2,1,b3]]);

GaussianElimination(M);

a:=spacecurve({[t, 3*t, -1*t, t = 0 .. 1]}, color = red, thickness = 2):

b:=spacecurve({[-5*t, -8*t, 2*t, t = 0 .. 1]}, color = blue, thickness = 2):

diagonalparallelogram:=spacecurve({[-4*t, -5*t, -1*t, t = 0 .. 1]}, color = black, thickness = 2):

c:=spacecurve({[0, 0, t, t = 0 .. 1]}, color = magenta, thickness = 2):

display(a,b,c,diagonalparallelogram);

1.7 definition of linearly independent - including motivating clicker question on span and connection to efficiency of span

Clicker questions in 1.7 and the theorem about l.i. equivalences in 1.7

Linearly independent and span checks:

li1:= Matrix([[1, 4, 7,0], [2, 5,8,0], [3, 6,9,0]]);

ReducedRowEchelonForm(li1);

span1:=Matrix([[1, 4, 7, b1], [2, 5, 8,b2], [3, 6, 9,b3]]);

GaussianElimination(span1);

Plotting - to check whether they are in the same plane:

a1:=spacecurve({[t, 2*t, 3*t, t = 0 .. 1]}, color = red, thickness = 2):

a2:=textplot3d([1, 2, 3, ` vector [1,2,3]`], color = black):

b1:=spacecurve({[4*t,5*t,6*t,t = 0 .. 1]}, color = green, thickness = 2):

b2:=textplot3d([4, 5, 6, ` vector [4,5,6]`], color = black):

c1:=spacecurve({[7*t, 8*t, 9*t, t = 0 .. 1]},color=magenta,thickness = 2):

c2:=textplot3d([7,8,9,`vector[7,8,9]`],color = black):

d1:=spacecurve({[0*t,0*t,0*t,t = 0 .. 1]},color=yellow,thickness = 2):

d2:=textplot3d([0,0,0,` vector [0,0,0]`], color = black):

display(a1, a2, b1, b2, c1, c2, d1, d2);

Linear Combination check of adding a vector that is outside the plane containing Vector([1,2,3]), Vector([4,5,6]), Vector([7,8,9]), ie b3+b1-2*b2 not equal to 0: Vector([5,7,10] as opposed to [5,7,9])

M:=Matrix([[1, 4, 7, 5], [2, 5, 8, 7], [3, 6, 9, 10]]);

ReducedRowEchelonForm(M);

Span check with additional vector:

span2:=Matrix([[1, 4, 7, 5,b1], [2, 5, 8,7,b2], [3, 6, 9,10,b3]]);

GaussianElimination(span2);

Linearly independent check with additional vector:

li2:= Matrix([[1, 4, 7, 5,0], [2, 5, 8,7,0], [3, 6, 9,10,0]]); ReducedRowEchelonForm(li2);

Removing Redundancy

li3:= Matrix([[1, 4, 5,0], [2, 5,7,0], [3, 6,10,0]]); ReducedRowEchelonForm(li3);

Adding the additional vector to the plot:

e1:=spacecurve({[5*t,7*t,10*t,t = 0 .. 1]},color=black,thickness = 2):

e2:=textplot3d([5,7,10,` vector [5,7,10]`], color = black):

display(a1, a2, b1, b2, c1, c2, d1, d2,e1,e2);

Roll Yaw Pitch Gimbal lock on Apollo 11.

dependence comic

Review via What's your span? comic.

Clicker questions in 1.3 and 1.5 # 1-3.

Clicker question in 1.4

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. 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).

1.5: vector parametrization equations of homogeneous and non-homogeneous equations. 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

Clicker question in 1.3 and 1.5 #4

discuss what happens when we correctly use GaussianElimination(s13n15extension) - write out the equation of the plane that the vectors span.

s13n15extension:=Matrix([[1,-5,b1],[3,-8,b2],[-1,2,b3]]);

GaussianElimination(s13n15extension);

How to express redundancy?

1.7 definition of linearly independent and connection to efficiency of span

In R^2: spans R^2 but not li, li but does not span R^2, li plus spans R^2.

span1:=Matrix([[1, 4, b1], [2, 5, b2], [3, 6, b3]]);

GaussianElimination(span1);

Comment on the span being b1-2b2+b3=0. Notice that Vector([7,8,9]) also satisfies this equation, and we can turn the plane they are in "head on" in Maple in order to see that no 2 lie on the same line but all are in the same plane:

a1:=spacecurve({[t, 2*t, 3*t, t = 0 .. 1]}, color = red, thickness = 2):

a2:=textplot3d([1, 2, 3, ` vector [1,2,3]`], color = black):

b1:=spacecurve({[4*t,5*t,6*t,t = 0 .. 1]}, color = green, thickness = 2):

b2:=textplot3d([4, 5, 6, ` vector [4,5,6]`], color = black):

c1:=spacecurve({[7*t, 8*t, 9*t, t = 0 .. 1]},color=magenta,thickness = 2):

c2:=textplot3d([7,8,9,`vector[7,8,9]`],color = black):

display(a1,a2,b1,b2,c1,c2);

What's your span? comic.

Begin 1.4. Ax via using weights from x for columns of A versus Ax via dot products of rows of A with x and Ax=b the same (using definition 1 of linear combinations of the columns) as the augmented matrix [A |b]. The matrix vector equation and the augmented matrix and the connection of mixing to span and linear combinations. Theorem 4 in 1.4

Review the algebra and geometry of eqs with 3 unknowns in R^3.

Clicker questions 1.1 and 1.2 #3 onwards

History of linear equations and the term "linear algebra" images, including the Babylonians 2x2 linear equations, the Chinese 3x3 column elimination method over 2000 years ago, Gauss' general method arising from geodesy and least squares methods for celestial computations, and Wilhelm Jordan's contributions.

Gauss quotation. Gauss was also involved in other linear algebra, including the history of vectors, another important "linear" object.

vectors, scalar mult and addition, Foxtrot vector addition comic by Bill Amend. November 14, 1999.

1.3 linear combinations and weights, vector equations and connection to 1.1 and 1.2 systems of equations and augmented matrix. linear combination language (addition and scalar multiplication of vectors).

c1*vector1 + c2*vector2_on_a_different_line is a plane via:

span1:=Matrix([[1, 4, b1], [2, 5, b2], [3, 6, b3]]);

GaussianElimination(span1);

Comment on the span being b1-2b2+b3=0. Notice that Vector([7,8,9]) also satisfies this equation

Clicker questions 1.1 and 1.2 #1. Mention solutions and a glossary on ASULearn.

Prepare to share your name, major(s)/minors/concentrations. Any questions?

Gaussian and Gauss-Jordan or reduced row echelon form in general: 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):

Review Drawing the line comic.

implicitplot3d({x+y+z=1, x+y+z=2}, x = -4 .. 4, y = -4 .. 4, z = - 4 .. 4)

Parametrize x+y+z=1.

with(plots): with(LinearAlgebra):

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);

Review the algebra and geometry of eqs with 3 unknowns in R^3.

Clicker questions 1.1 and 1.2 #2

Highlight equations with 3 unknowns with infinite solutions, one solution and no solutions in R

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 R

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);

In addition, do #4

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 last 2 slides

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);

Prove using geometry of lines that the number of solutions of a system with 2 equations and 2 unknowns is 0, 1 or infinite.

How to get to the main calendar page: google

Review Gaussian and Gauss-Jordan for 3 equations and 2 unknowns in R

Drawing the line comic. Solve the system x+y+z=1 and x+y+z=2 (0 solutions - 2 parallel planes)

implicitplot3d({x+y+z=1, x+y+z=2}, x = -4 .. 4, y = -4 .. 4, z = - 4 .. 4)

The following vocabulary is on the ASULearn glossary that I am experimenting with.

augmented matrix

coefficients

consistent

free

Gaussian elimination / row echelon form (in Maple GaussianElimination(M))

Gauss-Jordan elimination / reduced row echelon form (in Maple ReducedRowEchelonForm(M))

homogeneous system

implicitplot

implicitplot3d

linear system

line

parametrization

pivots

plane

row operations / elementary row operations

solutions

system of linear equations

unique