Test revisions

Share the final research presentations topic (name, major(s), concentrations/minors, research project idea, and whether you prefer to go 1st, 2nd or have no preference).

Reflection

Rubric for the final project

Review for test 3. Take questions on the study guide.

final research presentations

rubric for the final project

Evaluations.

Clicker questions---review of eigenvectors

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

Big picture discussion

final research presentations 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.)

sample project,

full guidelines

rubric for the final project

April was Mathematics Awareness Month on The Future of Prediction

Making a matrix disappear and then reappear 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...

Eigenfunction

Tacoma Narrows

Review: the algebra of eigenvectors and eigenvalues

Review trajectories from the glossary.

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--- eigenvector decomposition (5.6) part 2

Fill in examples on Terms for Test 3

Dynamical Systems and Eigenvectors remaining examples

final research presentations

Hamburger earmuffs and the pickle matrix

Clicker questions in Chapter 3 #9.

Test 2 corrections

Review: the algebra of eigenvectors and eigenvalues

Clicker questions in 5.1 #1-3

eigensheep comic

Eigenvector decomposition

Application: Foxes and Rabbits

Also revisit the black hole matrix.

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

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]]);

Eigenvalues and applications (5.1, 5.2 and 5.6)

Begin 5.1: the algebra of eigenvectors and eigenvalues and connect to geometry and Maple.

Eigenvalues of triangular matrices like shear matrix are on the diagonal-- characteristic equation.

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

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

Eigenvectors(M);

Eigenvector comic 1

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

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

Eigenvectors(M);

Application: Foxes and Rabbits

Overview of new material for test 2, study guide and take questions.

If space is the final frontier, then what's a subspace? 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.) Add to the terms. Two other examples.

Applications of 2.8

nullspace

Clicker questions in 2.8

Review 2.7 7 and 9

Review determinants LaTex Beamer slides

Mention google searches: application of determinants in physics application of determinants in economics application of determinants in chemistry application of determin ants in computer science Eight queens and determinants application of determinants in geology: volumetric strain

Clicker questions in Chapter 3 #4-8, 10

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.

Catalog description:

If space is the final frontier, then what's a subspace? subspace,

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 #7 and 8

Clicker questions in Chapter 3 #1-3

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);

glossary of terms

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.

Clicker 2.3 review

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

Clicker questions in 2.7 #1.

review linear transformations

Computer graphics demo [2.7] Examples 1-2

Clicker questions in 2.7 #2-6

Computer graphics demo [2.7] Examples 3-5

Keeping a car on a racetrack

Clicker questions in 2.7 #7

Go over Hill cipher and condition number

Clicker questions in 2.3 and Hill Cipher and Condition Number

Comic: associativity superpowers

Applications of 2.1-2.3: 1.8 (p. 62, 65, & 67-68), 1.9 (p. 70-75), and 2.7

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).

Mirror mirror comic and Sheared Sheap comic

general geometric transformations on R

In the process, review the unit circle

Test 1 corrections, day 1 slides.

Review 2.1 #21

Clicker in 2.1 continue with #8

In groups of 2-3 people, assume that A (square) has an inverse. What else can you say?

Theorem 8 in 2.3 [without linear transformations]: 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: Linear transformations in the cipher setting and finish 2.3 via the condition number.

Hill Cipher history

Maple file on Hill Cipher and Condition Number and PDF version

review of Hill cipher and condition number

Test 1

glossary of terms

multiply comic

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 continue with #7

Test 1 review part 2 and take questions on the study guide

Review matrix addition, scalar multiplication and transpose and matrix multiplication.

matrix algebra. AB not BA...

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:

Finish Test 1 review part 1

Begin Chapter 2:

via Clicker questions in 2.1 1-4

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 in LaTeX.

matrix multiplication

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

Roll Yaw Pitch Gimbal lock on Apollo 11.

clicker review questions

Maple commands

Test 1 review part 1

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);

1.5: 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 #5

Clicker question to motivate 1.7

How to express redundancy?

1.7 definition of linearly independent and connection to efficiency of span

Fill in glossary

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

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

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);

Replace with [7, 8, 10] which is not in the span.

Clicker questions in 1.3 and 1.5 # 1, 2

What's your span? comic.

Clicker questions in 1.3 and 1.5 # 3-4

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. The matrix vector equation and the augmented matrix and the connection of mixing to span and linear combinations.

Theorem 4 in 1.4

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).

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.

Glossary 2: More Terms for Test 1

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

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);

Replace with [7, 8, 10] which is not in the span.

Clicker questions in 1.3 and 1.5 # 1, 2

Engagement with the the i-clickers

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

Clicker on 3eqs 2 vars

Clicker questions 1.1 and 1.2 #1.

Mention engagement, solutions and a glossary on ASULearn.

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):

Parametrize x+y+z=1.

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

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);

Ex4a:=Matrix([[1,3,4,k],[2,8,9,0],[10,10,10,5],[5,5,5,5]]);

GaussianElimination(Ex4);

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 terms in 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 few 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

MyMathLab

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)