2240 class highlights

  • Tues May 1
    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
  • Thur Apr 27
    final research presentations
    evaluations
    test 2 corrections
    presentation session,
    course overview
    sample project,
    full guidelines, rubric for the final project

  • Tues Apr 25 Test 2
  • Thur Apr 20 Math Awareness Month
    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
  • Thur Apr 13
    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

  • Tues Apr 11
    Clicker questions--- eigenvector decomposition (5.6) part 2
    reviewing
    Fill in examples on Terms for Exam 2
    Dynamical Systems and Eigenvectors
  • Thur Apr 6
    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?

  • Tues Apr 4 eigensheep comic and review algebra of eigenvalues and eigenvectors
    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
  • Thur Mar 30 Take questions on determinants.
    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]])

  • Tues Mar 28 Take questions on determinants.
    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
  • Thur Mar 23
    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?

  • Tues Mar 21
    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.
  • Thur Mar 9 rotation matrix and 6.1
    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

  • Tues Mar 7 Test 1
  • Thur Mar 2
    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

  • Tues Feb 28
    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 R2 [1.8, 1.9]
    In the process, review the unit circle
    Computer graphics demo [2.7] Examples 1-2
    Clicker questions in 2.7 #1-3
  • Thur Feb 23
    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).

  • Tues Feb 21
    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
  • Thur Feb 16
    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?

  • Tues Feb 14
    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)
  • Thur Feb 9 Review
    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

  • Tues Feb 7
    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
  • Thur Feb 2
    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

  • Tues Jan 31 Review vectors, addition, scalar multiplication, linear combinations and span of them
    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
  • Thur Jan 26
    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

  • Tues Jan 24
    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
  • Thur Jan 19 Turn in hw.
    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 R2.
    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 R3, and the corresponding geometry, as we review new terminology and glossary of terms

  • Tues Jan 17 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.

    How to get to the main calendar page: google Dr. Sarah / click on webpage / then 2240. Discuss webpages, hw and Polya's How to Solve it
    Vocabulary/terms/ASULearn glossary