As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality -Albert Einstein (1879-1955)
  • Syllabus

     

  • Instructor's contact info: Dr. Al Parker; email: parker@math.montana.edu; phone: 994-5145; office: EPS304; office hours and schedule.

     

  • Online Resources:

     

  • Exams:

     

  • Quizzes:
    • 4/29 In class Quiz 11 on complex eigenpairs and difference equations, §8.3. Solutions
    • DUE: 4/27, Take Home Quiz 9 on Chapters 1-6 using MATLAB. This lab may be useful to get you going. Solutions
    • 4/22 In class Quiz 10 on solving ODEs with complex eigenpairs, §6.3. Solutions
    • 4/13 In class Quiz 8 on §6.1. Solutions
    • DUE: 3/27 Take Home Quiz 7 on the Fredholm Alternative and projections. Solutions: page 1, page 2.
    • 3/20 In class Quiz 6 on the dimensions of the 4 fundamental spaces. Solutions
    • 3/4 In class Quiz 5 on finding solutions, the domain and range of rectangular systems, and vector spaces. Solutions
    • 2/13 In class Quiz 4 on §2.5 Matrix inverses. Solutions
    • 2/4 In class Quiz 3 on §2.1-2.2 Gauss Elimination, particular solutions and null solutions. Solutions
    • 1/28 In class Quiz 2 on dot products, Cauchy Schwarz, Triangle Inequality. Solutions
    • DUE: 1/23 Take home Quiz 1. Solutions

     

  • Course Schedule:
    • 5/1 Course Review. VIDEO of Strang's course review.
    • 4/29 §8.3 Complex eigenvalues in difference equation models
    • 4/27 §8.3 Using eigenvalues in ecology and economics: describing the long term behavior of a difference equation model of a predator-prey system. HW in §8.3: 1-3, 7

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    • 4/24 §6.1, 6.3, 6.4 How to check ODE solutions. A summary of the properties of eigenvalues and eigenvectors. Symmetric matrices are special matrices because (incredibly) they are orthogonally diagonlizable and have only real eigenpairs. VIDEO of Strang's lecture on symmetric matrices. Read §8.3; HW in §6.1: 16, 17; HW in §6.4: 2-5, 21, 22
    • 4/21 §6.3 Using complex eigenpairs to solve systems of ODEs.
    • 4/19 §6.3 Using real eigenpairs to solve systems of ODEs. VIDEO of Strang's lecture re: solving ODEs with eigen-pairs. HW in §6.3: 1, 2, 4, 5, 8

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    • 4/17 §6.2 When is diagonalization possible? ANS: When Algebraic = geometric multiplicity
    • 4/15 §6.1, 6.2 The eigenvector(s) associated with a zero eigenvalue of A is/are another way to describe a basis for N(A); How to diagonalize or find the eigendecomposition of a matrix VIDEO of Strang's lecture on Diagonalization. HW in §6.2: 1-4, 6, 7, 11, 12, 14-17, 20, 26, 27, 30, 33
    • 4/13 §6.1 Complex eigenvalues and eigenvectors always come in conjugate pairs.

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    • 4/10 §6.1 How to find eigenpairs of an n x n matrix A using the n-th order characteristic polynomial; Read §6.2-6.4; HW in §6.1: 2-6, 9, 10, 12-15, 21, 24, 27-29, 34
    • 4/8 §6.1, 6.2 The n linearly independent eigenvectors of an nxn mattrix A form a basis for n-D space, and allow an eigendecoomposition of A = S*Lambda*S^(-1).
    • 4/6 §6.1 Eigenproblem, eigenvalue, eigenvector, eigenpair. Eigenpairs describe how the n x n matrix A stretches, squishes, flips and/or rotates vectors in n-D space. VIDEO of Strang's lecture on the eigen-problem. Read §6.1; HW in §6.1: 32

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    • 4/3 HAPPY SPRING! NO CLASS.
    • 4/1 Exam 2
    • 3/30 Review of Chapters 1-5

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    • 3/27 §4.3 The method of least squares. VIDEO of Strang's lecture on least squares. HW in §4.3: 1-3, 5-7, 9, 10, 17, 18, 20, 22
    • 3/25 §4.2 Projections and projection matrices. VIDEO of Strang's lecture on projections. HW in §4.2: 1-3, 11-14, 16-18
    • 3/23 §4.1 The Fredholm Alternative. Read problem #7 in §4.1; Read §4.2-4.3

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    • 3/20 §3.1-3.6 Reviewing chapter 3 like Strang does it: §3.1: vector spaces, C(A); §3.2: N(A), ref(A), rref(A); §3.3: The rank of a matrix rank(A) = dim(C(A)) = dim(R(A)); §3.4: particular + null solutions; §3.5: linear independence, spans, bases, dimension, R(A); §3.6: Four Fundamental Spaces, N(A'), Dimension Theorem.
    • 3/18 §3.2, 3.6, 4.1 Finding the basis for N(A). The dimensions of the 4 fundamental subspaces. The Orthogonal Decomposition Theorem. VIDEO of Strang's 4 fundamental subspace lecture. Read §4.2-4.3; HW in §3.6: 1-3, 6-8, 10, 11, 13, 17, 24; HW in §4.1: 1, 2, 3abc, 5, 7, 10-12, 18, 19, 21
    • 3/16 §3.2, 3.4 Null space of a matrix N(A) is the space of all null (homogeneous) solutions. VIDEO of Strang's column space and nullspace lecture. Read §3.6-4.1; HW in §3.2: 1, 2, 4-10, 13-18, 29, 34; HW in §3.4: 1-4, 8, 13, 16-19, 21, 23-30

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    • 3/9 - 3/13 SPRING BREAK!!

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    • 3/6 §4.3 Extra Credit Group Work: applying the method of least squares to data collected from your classmates.
    • 3/4 §3.5,3.6 Row space of a matrix R(A), how to find a basis of the column and row spaces using GE and the ref(A). The Dimension Theorem: dim(C(A)) = dim(R(A)). Proof: The dim of each is the number of pivots.
    • 3/2 §3.5 Linear independence, the span of a set of vectors, basis for a vector space, dimension of a vector space, how to find a basis of the column space using GE and the ref(A). VIDEO of Strang's lecture on vector space bases and dimension. Read §3.2 and 3.4; HW in §3.3: 1, 2a; HW in §3.5: 1, 2, 5, 6, 9, 11, 12, 13, 17, 21, 23, 24

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    • 2/27 §3.1 Vector spaces, vector subspaces, checking closure with respect to addition (A1) and closure with respect to scalar multiplication (M1). Read §3.5; HW in §3.1: 1, 2, 4-6, 10-16, 19, 20, 22, 26-29; HW in §3.4: Give the domain and range of the coefficient matrix A in 1, 2, 4, 21 and 30
    • 2/25 §3.1-3.4 Solving rectangular systems. Column space of a matrix, C(A). HW in §3.1: Solve Ax=0, Bx = 0, Cx = 0 in 19; HW in §3.2: Solve Ax=0 and Bx = 0 in 1ab and 5ab; HW in §3.3: For the 3x4 R and the 3x3 R solve Rx = 0 in 7; HW in §3.4: Solve Ax = b in 1-2; do 4, 21 and 30
    • 2/23 §3.1 Viewing A as a function with a domain and range=column space. The column space is a vector spaces. Read §3.3; HW in §3.1: Give the domain and range of A, B and C in 19; HW in §3.2: Give the domain and range of A and B in 1ab and 5ab; HW in §3.3: Give the domain and range of the 3x4 R and the 3x3 R in 7.

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    • 2/20 Exam 1
    • 2/18 Review Chapters 1, 2 and 5
    • 2/16 PRESIDENTS DAY

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    • 2/13 §5.1 Properties of determinants. VIDEO of Strang's lecture about determinant properties. Read §3.1; HW in §5.1: 1-3, 12
    • 2/11 §5.2 Using Laplace Cofactors to find the determinant of an n x n matrix. VIDEO of Strang's lecture about determinants. HW in §5.2: Use cofactors to find the determinants in 1-3, find det(B) in 11, find det(A) in 12, 13a, 15b, 20, 21
    • 2/9 §2.5, 2.7 Inverse matrix properties; symmetric matrices; the transpose. HW in §2.5: 6, 31; HW in §2.7: 1-5, 7, 16, 17a, 18a, 19;

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    • 2/6 §2.6 How to find an inverse matrix: Gauss-Jordan elimination. Reduced row echelon form. HW in §2.5: 1-5, 10-13, 16, 18, 19, 22-25, 27-30, 32;
    • 2/4 §3.4 If a system has 1 solution, can that solution still be written as a particular + a null (or homogeneous) solution? YES! But the null solution in this case is boring (i.e. x_n = 0).
    • 2/2 §2.5, 5.1 Invertibility and determinants? The determinant is the product of the pivots. Row echelon form. Read §5.1-5.2; HW in §2.4 to practice matrix operations: (already assigned) 1,3,5,6; (more problems) 14, 20-22, 26, 27, 32; HW in §5.1: 9, 13-15, 18, 19, 22, 23, 27

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    • 1/30 §2.5 What exactly is an inverse matrix? The identity matrix. VIDEO of Strang's inverse matrix lecture. Read §2.7;
    • 1/28 §2.2, 3.4 How to write a general solution as a particular + a null (or homogeneous) solution. When does Gauss Elimination (GE) fail to give a single solution? Answers: When each variable does not have a pivot. This happens when one of the following occur: when a row switch is required; when there are no solutions; when there are an infinite number of solutions. Read §2.4-2.5; HW in §3.4: 3, 4, 21 (write the general or complete solution in each case as a particular solution + a null solution)
    • 1/26 §2.1-2.2 Gauss Elimination, augmented matrix. VIDEO of Strang's lecture on elimination. HW in §2.1: 15-18, 26-28 HW in §2.2 (some of these assigned earlier, but now use GE to solve): 1-3, 5, 12-15, 17, 19, 21, 25, 27; HW in §2.3: 16, 17, 24, 25, 27

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    • 1/23 §1.2-1.3, 2.1 The Cauchy-Schwarz and triangle inequalties; the row picture when solving a 3-D system. Read §2.3; HW on §1.2: 2; HW on §1.3: 1, 2, 4; HW in §2.1: 4-8
    • 1/21 §1.2-1.3, 2.4 Dot product, matrix multiplication, vector lengths (norms), orientation via the dot product. HW from §1.2: 1, 3, 4, 7, 11, 13, 14, 16, 25; HW from §2.1: 9-14; HW from §2.4: 1,3,5,6
    • 1/19 MARTIN LUTHER KING DAY

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    • 1/16 §1.1,2.1 Row picture and column picture, vectors, matrices and linear combinations. Read §1.2-1.3, 2.4; HW from §1.1: 1-8, 10-13, 15-26; HW from §2.1: 1-2.
    • 1/14 §1.1-2.2 Introducton to systems of linear equations: how to find solutions; how many solutions; how to describe the solutions. VIDEO of Strang's introductory lecture. Read §1.1, 2.1-2.2; HW from §2.2: 1-3, 5; solve the 3x3 systems in 12 and 13. 

     

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