applications of linear algebra

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).

rubric for the final project

MathSciNet Hill cipher. Leontief. Search within matrix/matrices

evaluations

final research presentations

evaluations

test 2 corrections

presentation session,

course overview

sample project,

full guidelines, rubric for the final project

uncover the mystery of inverse(P).A.P=?, Diagonalization and apply to computer graphics

Applications to mathematical physics, quantum chemistry..., Eigenfunction, Tacoma Narrows,

full guidelines and rubric for the final project

evaluations

Test 2 review, topics to study, Test 1 review

Clicker questions---review of eigenvectors

reviewing, course goals

April is Mathematics Awareness Month

THE $25,000,000,000 EIGENVECTOR by Kurt Bryan and Tanya Leise

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

presentation session, 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.)

Hamburger earmuffs and the pickle matrix

sample project,

full guidelines

rubric for the final project

Big picture discussion

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

reviewing

Fill in examples on Terms for Exam 2

Dynamical Systems and Eigenvectors

Clicker questions in 5.1#1-3

Review first example on Dynamical Systems and Eigenvectors

Clicker questions on eigenvector decomposition (5.6) part 1#2-4 [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 basis?

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

Also revisit the black hole matrix.

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

Compare with Dynamical Systems and Eigenvectors first example

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

Eigenvector comic 2

Clicker questions in Chapter 3 9

basis, null space and column space

clicker in 2.8#2-3

algebra of eigenvalues and eigenvectors and connect to geometry

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

mapping course goals to the text

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.

nullspace

clicker in 2.8 1

Review determinants LaTex Beamer slides via Clicker questions in Chapter 3 #10, 4-8

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.

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

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

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 determin ants in computer science Eight queens and determinants application of determinants in geology: volumetric strain

3.3 p. 180-181:

Begin 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.

Review linear transformations of the plane, including homogeneous coordinates

glossary of terms rotation matrix and 6.1

Application of 2.7 and 6.1: Keeping a car on a racetrack

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, 8 and 9

Clicker questions in Chapter 3 #1-3

Review linear transformations of the plane, including homogeneous coordinates

Clicker questions in 2.7 #4-6

Computer graphics demo [2.7] Examples 3-5

Test 1 review and take questions

Time to study for the test or work on 1.9 and 2.7 hw

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

Glossary of terms

general geometric transformations on R

In the process, review the unit circle

Computer graphics demo [2.7] Examples 1-2

Clicker questions in 2.7 #1-3

Clicker questions in 2.2 #4-5

Clicker questions in 2.3 and Hill Cipher and Condition Number

review of Hill cipher and condition number

What makes a matrix invertible

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).

Clicker questions in 2.2 #1-3

2.1 #23: Assume CA=I_nxn. A doesn't have to be square. 3x2 matrix A.

last slide for advice from students

2.2 #21: Explain why the columns of an nxn matrix A are linearly independent when A is invertible.

problematic reasoning: If the 2 columns of A are multiples the determinant will be 0

incomplete reasoning: the columns of A are li because Ax=0 has only the trivial solution when A is invertible (why?).

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)

Introduction to Linear Maps

Hill Cipher history

Maple file on Hill Cipher and Condition Number and PDF version

review of Hill cipher and condition number

Comic: associativity superpowers

Review 2.1 #23 (multiplicative argument and then pivot argument)

Steps, The Science of Successful Learning, learn something new

Review 2.2 Algebra: Inverse of a matrix and 2.1 #21.

Clicker in 2.1 and 2.2 continued: #7 onward

Show that if the columns of a square nxn matrix A span the entire R^n, then A is invertible.

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

Continue matrix algebra including addition and scalar multiplication.

Then 2.1 question, 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).

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.

comic. Find the identity of superman

Applications of multiplication and the inverse (if it exists)

1.7 definition of linearly independent

dependence comic

Review Maple commands Maple file

Review 1.1, 1.2, 1.3, 1.4, 1.5, 1.7

clicker review questions 4-9

Begin Chapter 2:

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

glossary for 2.1-2.3 and matrix algebra

Clicker question to motivate 1.7

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.

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

Roll Yaw Pitch Gimbal lock on Apollo 11.

Maple commands Maple file

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).

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

What's your span? comic.

Clicker questions in 1.3 and 1.5 # 3-4

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);

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);

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

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

Collect hw. Go over the glossary on ASULearn, solutions, hints, and advice from the last run of the class.

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

Clicker questions in 1.1 and 1.2 continued

Engagement with the the i-clickers

Clicker questions 1.1 and 1.2 #1.

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

Clicker on 3eqs 2 vars

Mention where to get help, 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):

Drawing the line comic.

Parametrize x+y+z=1.

with(plots): with(LinearAlgebra):

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

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 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

Vocabulary/terms/ASULearn glossary