2240 class highlights

  • Fri Jun 29 Finish final presentations

  • Thur Jun 28 Begin final presentations

  • Wed Jun 27
    Discuss the final project presentations. Divide up the sessions based on topics.
    Work on project and/or test revisions.
    Evaluations for those who missed Monday.

  • Tues Jun 26 Test 2.

  • Mon Jun 25
    second review activity
    success
    Test 2 review, topics to study
    Practice test, problem sets, hw problems, clickers, study guide topics, glossaries. Solutions exist for you to compare and learn from but be sure to try them on your own and make sure you can discuss the concepts and do the problems (linearly +) independently!
    final project presentations
    evaluations

  • Fri Jun 22
    Clicker in Chapter 5 #14-18
    course goals
    uncover the mystery of inverse(P).A.P=?, Diagonalization and apply to computer graphics-
    Applications to mathematical physics, quantum chemistry..., Standing wave, Eigenfunction, Tacoma Narrows
    MathSciNet Hill cipher. Leontief. Search within matrix/matrices.
    Google Scholar: eigenvalue in mathematics education research
    full guidelines, topics and sample projects introduction to LaTex.
    review activity

  • Thur Jun 21
    Clicker in Chapter 5 #8-13
    Continue Dynamical Systems and Eigenvectors
    comic http://brownsharpie.courtneygibbons.org/comic/guest-artist-little-pete-is-emo/
    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.
    Eigenfeet, eigenfaces, eigenlinguistics
    presentation session http://hosted.jalt.org/pansig/2005/HTML/Bayne.htm, final research presentations
    Hamburger earmuffs and the pickle matrix
    full guidelines, topics and sample projects
    rubric for the final project

  • Wed Jun 20
    Review algebra of eigenvalues and eigenvectors and Eigenvector decomposition
    Clicker in Chapter 5 #2-5 Then continue and highlight predator prey, predator predator or cooperative systems (where cooperation leads to sustainability) and #6-7.
    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]])

  • Tues Jun 19
    Take questions on 2.8. Engagement
    algebra of eigenvalues and eigenvectors and connect to geometry
    eigensheep comic
    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);
    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 in Chapter 5 #1
    Compare with Dynamical Systems and Eigenvectors first example

  • Mon Jun 18
    Clicker questions in chapter 3 #10
    If space is the final frontier, then what's a subspace? subspace Paramount and CBS, basis, null space and column space
    nullspace null=me!
    clickers in 2.8 1-3
    algebra of eigenvalues and eigenvectors and connect to geometry
  • Fri Jun 15
    Clicker questions in 2.7 #7-9
    Review Laplace expansion of the determinant LaTex Beamer slides
    The determinator comic, which has lots of 0s, review row operations and determinants
    The relationship of row operations to the geometry of determinants - shear matrices preserve area, volume.
    Clicker questions in chapter 3#4-9
    If space is the final frontier, then what's a subspace? subspace Paramount and CBS
    Engagement and exam corrections

  • Thur Jun 14 Test 1

  • Wed Jun 13
    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);
    LaTex Beamer slides
    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
    Moving activity: Glossary matchup
    review slides, study guide, sample partial test

  • Tues Jun 12
    Review linear transformations of the plane, including homogeneous coordinates
    Computer graphics demo [2.7] Example 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
    Clicker questions in 2.7 #4-6
    Clicker questions in chapter 3#1-3

  • Mon Jun 11 List relevant examples and course overview
    Clicker in 2.1-2.3 #20-22. Discuss problem set and create a video.
    Text 2 material: Glossary of terms and more glossary, clickers Linear transformations continued.
    Moving activity: Each odd person moves +4 (mod class size).
    Guess the transformation. 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, 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 http://digmi.org/tag/fun/page/2/ and Sheared Sheap comic from our book
    general geometric transformations on R2 [1.8, 1.9]
    In the process, review the unit circle
    Computer graphics demo [2.7] Example 1
  • Fri Jun 8
    Show that if the columns of a square nxn matrix A span the entire R^n, then A is invertible.
    Clicker in 2.1-2.3 #10-12
    2.1 #23: Assume CA=I_nxn. A doesn't have to be square. 3x2 matrix A.
    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
    List relevant examples and course overview
    -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
    Clicker in 2.1-2.3 #13-19

  • Thur Jun 7
    Comic: associativity superpowers
    2.2 Algebra: Inverse of a matrix.
    Steps, The Science of Successful Learning, learn something new
    Divide up using cut up comics Clicker in 2.1-2.3 #5-9
    Applications of multiplication and the inverse (if it exists)
    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.
    Pivot and matrix multipliction arguments: A invertible Ax=b solutions, Ax=0 solutions.

  • Wed Jun 6
    Maple commands Maple file file, Clicker question in 1.3, 1.4, 1.5, 1.7 #18-
    Begin Chapter 2:
    Image 1   Image 2   Image 3   Image 4   Image 5   Image 6   Image 7.
    glossary for 2.1-2.3
    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).
    twobytwo := Matrix([[a, b], [c, d]]);
    MatrixInverse(twobytwo);
    MatrixInverse(twobytwo).twobytwo
    simplify(%)
    comic. Find the identity of superman
    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.

  • Tues Jun 5
    Review 1.5 and 1.3 and 1.7 vector and matrix equations, Theorem in 1.7
    Clicker question in 1.3, 1.4, 1.5, 1.7 #8-10
    dependence comic
    Roll Yaw Pitch Gimbal lock on Apollo 11.
    Break up via Random sequence generators and Review 1.1, 1.2, 1.3, 1.4, 1.5, 1.7
    Clicker question in 1.3, 1.4, 1.5, 1.7 #11-

  • Mon Jun 4 Discuss problem sets. Review Clicker in 1.3-1.7 # 5 in Maple
    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);
    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({[-4*t, -5*t, 3*t, t = 0 .. 1]}, color = magenta, thickness = 2):
    display(a,b,c,diagonalparallelogram);
    Modified the diagonal of the parallelogram by changing the last coordinate to get a vector out of the plane.
    Review 1.3 and 1.4, and theorem 4 in 1.4.
    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.
    Clicker in 1.3-1.7 # 6 and #7 to motivate 1.7
    How to express redundancy?
    1.3 and 1.7 vector and matrix equations
    In R^2: spans R^2 but not li, li but does not span R^2, li plus spans R^2.
    Theorem in 1.7
  • Fri Jun 1 Review vectors, addition, scalar multiplication, linear combinations and span of them, and movie visualizations: span2dmovie, spand3dmovie.
    What's your span? comic
    Clicker questions in 1.3, 1.4, 1.5, 1.7 # 3
    Maple
    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.
    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
    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).
    Clicker in 1.3, 1.4, 1.5, 1.7 #4-5

  • Thur May 31
    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 was also involved in other linear algebra, including the history of vectors, another important "linear" object.
    Advice from previous students
    2240 engagement clicker questions 1.1 and 1.2 continued #6 onward
    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).
    Begin clicker in 1.3-1.7

  • Wed May 30 Turn in hw. Gaussian and Gauss-Jordan for 3 equations and 2 unknowns in R2.
    Engagement with the i-clickers (Think, Pair up, Share, Review and Add), where to get help, solutions and glossary on ASULearn. Exam 1 questions. Clicker in 1.1 and 1.2 #1.
    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. Maple
    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(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 clicker questions 1.1 and 1.2 continued

  • Tues May 29 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);
    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 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);
    Prove using geometry of lines that the number of solutions of a system with 2 equations and 2 unknowns is 0, 1 or infinite.

    Review Gaussian and Gauss-Jordan for 3 equations and 2 unknowns in R2.

    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)