• Thur Dec 4
    Reflection
    final research presentations, full guidelines and sample project.
    Clickers on final research topic
    Share the final research presentations topic with the rest of the class (name, major(s), concentrations/minors, research project idea, and whether you prefer to go 1st, 2nd or have no preference).
    Informal evaluation while I check in about the projects, and then formal evaluations.

  • Tues Dec 2
    April was mathematics awareness month - the theme was magic, mystery and mathematics. 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 Rn)-diagonalizability. [We can uncover the mystery and apply this to computer graphics].
    Applications to mathematical physics, quantum chemistry...
    Review final research presentations
    Test 3 revisions
  • Tues Nov 25 Test 3
  • Thur Nov 20
    final research presentations slide 2 and 3. 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
    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
    Big picture discussion
    Clicker survey questions and consent form

    Review for test 3 and take questions on the study guide, material or final research presentations

  • Tues Nov 18
    Clicker questions--- eigenvector decomposition (5.6) part 2
    Clicker questions---review of eigenvectors
    final research presentations page 1
    Hamburger earmuffs and the pickle matrix
  • Thur Nov 13
    Write down any eigenvector or eigenvalue equations we have focused on.
    Two mantras.
    Review reflection matrix via pictures. A few inputs. Where is the output? Is the vector an eigenvector?
    Geometry of Eigenvectors examples one and two and compare with Maple
    Ex1:=Matrix([[0,1],[1,0]]);
    Ex2:=Matrix([[0,1],[-1,0]]);
    Geometry of Eigenvectors examples three and four and compare with Maple
    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?

  • Tues Nov 11
    Eigenvector comic 1 Review Eigenvalues and Eigenvectors and the Eigenvector decomposition
    Why we use the eigenvector decomposition versus high powers of A for longterm behavior (reliability)
    Clicker questions on eigenvector decomposition (5.6) part 1#2
    Compare with Dynamical Systems and Eigenvectors
    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: Ex1:=Matrix([[0,1],[1,0]]);
    Eigenvalues(Ex1);
    Eigenvectors(Ex1);
  • Thur Nov 6
    Clicker questions in 5.1#1-3
    Matrix([[2,1],[1,2]])
    M := Matrix([[2,1],[1,2]]);
    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]]);
    Application: Foxes and Rabbits
    Compare with Dynamical Systems and Eigenvectors first example
    Clicker questions on eigenvector decomposition (5.6) part 1#1

  • Tues Nov 4
    Clicker questions in 2.8
    Begin 5.1: the algebra of eigenvectors and eigenvalues, and connect to geometry and Maple.
    Also revisit the black hole matrix. Eigenvalues and eigenvectors via the algebra as well as the geometry.
  • Thur Oct 30
    Clicker questions in Chapter 3 9
    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
  • Tues Oct 28
    Clicker questions in Chapter 3 10
    Questions on 3.1 or 3.2.
    Graphic on steps
    Catalog description: A study of vectors, matrices and linear transformations, principally in two and three dimensions, including treatments of systems of linear equations, determinants, and eigenvalues.
    Clicker questions in Chapter 3 3
    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.
    Clicker questions in Chapter 3 4-7
  • Thur Oct 23 Test 2
  • Tues Oct 21
    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

    Overview of new material for test 2 and take questions.
  • Tues Oct 14
    Review Problem Set 3 #1
    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
  • Thur Oct 9
    Clicker question
    general geometric transformations on R2 [1.8, 1.9], including homogeneous coordinates
    linear transformation comic
    Computer graphics demo [2.7] Examples 1 and 2
    Clicker questions in 2.7 #2 -4
    Linear transformations of 3-space: Computer graphics demo [2.7] Examples 3-5

  • Tues Oct 7
    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
    Continue computer graphics and linear transformations (1.8, 1.9, 2.3 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 R2 [1.8, 1.9]
    In the process, review the unit circle
  • Thur Oct 2
    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.
    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

    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

  • Tues Sep 30
    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.
    Applications of 2.1-2.3: 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

    Hill Cipher history
    Maple file on Hill Cipher and Condition Number and PDF version
  • Thur Sep 25 Review transpose of a matrix, 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 corrections

  • Tues Sep 23 Test 1
  • Thur Sep 18
    Test 1 review part 2
    Take review questions for test 1. Continue Chapter 2.
    Continue 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

  • Tues Sep 16
    Test 1 review part 1
    Begin Chapter 2:
    Continue via Clicker questions in 2.1 1-4

    Image 1   Image 2   Image 3   Image 4   Image 5   Image 6   Image 7.
  • Thur Sep 11
    dependence comic
    Clicker review questions

  • Tues Sep 9 Collect homework and take questions on 1.4 or 1.5.
    How to express redundancy?
    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
    In R^2: spans R^2 but not li, li but does not span R^2
    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.
  • Thur Sep 4
    What's your span? comic
    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 and #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,d);

  • Tues Sep 2
    Clicker question on Problem Set 1, collect 1.3. Review language from Thursday:
    Clicker questions in 1.3 and 1.5 # 1-3.
    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
  • Thur Aug 28 Collect problem set 1.
    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. 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, 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);
    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

  • Tues Aug 26
    Collect homework
    Review the algebra and geometry of eqs with 3 unknowns in R^3.
    Clicker questions 1.1 and 1.2 #2 onward
  • Thur Aug 21 Turn in hw. Register the i-clickers.
    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 for 3 equations and 2 unknowns in R2.

    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)
    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);
    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 words.

  • Tues Aug 19 UTAustinXLinearAlgebra.mov. Manga comic
    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 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.

    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)

    How to get to the main calendar page: google Dr. Sarah / click on webpage / then 2240

    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