- 1.2. Matrices and Gaussian Elimination Suppose you need to solve the following system of
- 10856. Proposed by Andrei Jorza, "Moise Nicoara" High School, Arad, Romania. Find all bounded convex polyhedra such that no three faces have the same number of edges.
- 1.1. Introduction to Linear Systems Algebra deals mainly with equations of one variable.
- C. HECKMAN MAT 265 Solutions Test 2
- A New Proof Of The Independence Ratio Of TriangleFree Cubic Graphs
- 11377. Proposed by Christopher Hillar, Texas A&M University, College Station, TX, and Lionel Levine, Massachusetts Institute of Technology, Cambridge, MA. Given a monic polynomial p of degree n with complex
- 11406. Proposed by A. A. Dzhumadil'daeva, Almaty, Republics Physics and Mathematics School, Almaty, Kazakhstan. Let n!! denote the product of all positive integers not greter than n and congruent to n mod 2,
- Rook Polynomials Christopher Carl Heckman
- 11208. Proposed by Li Zhou, Polk Community College, Winter Haven, FL. (Abbreviated) Let Mn be the stage-n Menger sponge, where M0 is the unit cube. The chromatic number of a surface is the minimum
- 11258. Proposed by Manuel Kauers, Research Institute for Symbolic Computation, Johannes Kepler Uni-versity. Let Fn denote the nth Fibonacci number, and let i denote
- Series Convergence/Divergence Flow Chart TEST FOR DIVERGENCE
- Matrix Operations on a TI-83 Graphing Calculator Christopher Carl Heckman
- Solutions to Test 3 Problems are identified as (form, problem number,
- 5.1. Orthogonal Vectors in Rn When dealing with vectors in R2
- Sixty-Seventh Annual William Lowell Putnam Mathematical Competition (2006) (A1) Find the volume of the region of points (x, y, z) such that
- Matrix Applications: Markov Chains and Game Theory Christopher Carl Heckman
- 11333. Proposed by Pablo Fernandez Refolio, Universidad Autonoma de Madrid, Madrid, Spain. Show that Solution by Christopher Carl Heckman, Arizona State University, Tempe, AZ: Let
- 6.1. Eigenvalues1 and Eigenvectors2
- 10846. Proposed by Jozef Przytycki, The George Washington University, Washington, DC. A lattice simple closed curve in the plane is a simple closed curve all of whose points have an integer coordinate. The area
- 7.3 Coordinates and Change of Basis If B = {v1, v2, . . . , vk} is a basis for a subspace W, then
- 1.4. Matrix Operations 1.4. Matrix Operations
- 10900. Proposed by Gordon Rice, Davis, CA. It is clear from the law of cosines that every angle that occurs in a triangle with integer sides has a rational cosine. Is the converse true? Does every angle between 0 and
- Matrix Operations on a Casio Graphing Calculator Christopher Carl Heckman
- AMM 11218: C. C. Heckman's Solution 1 11218. Proposed by Gary Gordon, Lafayette College, Easton, PA. Consider the following algorithm, which
- C. HECKMAN TEST 1A SOLUTIONS 170
- C. HECKMAN MAT 271 SOLUTIONS Test 3
- 1.5. Inverses of Matrices* There are actually two things we need to do, to make
- 11394. Proposed by K. S. Bhanu, Institute of Science, Nagpur, India, and M. N. Deshpande, Nagpur, India. A fair coin is tossed n times, with n 2. Let R be the resulting number of runs of the same face, and X
- 11515. Proposed by Estelle L. Basor, American Institute of Mathematics, Palo Alto, CA, Steven N. Evans, University of California, Berkeley, CA, and Kent E. Morrison, California Polytechnic State University, San
- Independent Sets In Triangle-Free Cubic Planar Graphs POST-REFEREED DRAFT #2
- On the Tightness of the 5/14 Independence Ratio [Draft #2] Christopher Carl Heckman
- Using Graphing Calculators To Evaluate Riemann Sums Christopher Carl Heckman
- Linear Programming: Beyond 4.2 (The Simplex Method) Christopher Carl Heckman
- 10683. Proposed by Harry Tamvakis, University of Pennsylvania, Philadelphia, PA. Let a1, . . . , an be a sequence of nonzero real numbers, exactly p of which are positive. Characterize the pairs (n, p) such that
- 10966. Proposed by Jean-Marie De Koninck, Universite Laval, Quebec, Canada. (a) Let denote the Euler function, and let (n) = p|n p, with (1) = 1. Prove that there are exactly
- 11187. Proposed by Li Zhou, Polk Community College, Winter Haven, FL. Find a closed formula for the number of ways to tile a 4 by n rectangle with 1 by 2 dominoes.
- 11196. Proposed by Mohammad Hossein Mehrabi, Iran University of Science and Technology, Tehran, Iran. Let A and B be real n n matrices. Show that if AB -BA is invertible and A2
- 11209. Proposed by Alexander Dubinov and Irina Dubinova, Russian Federal Nuclear Center, Sarov, Russia. Consider the following system of n equations in n positive unknowns x1, . . . , xn and n positive given numbers
- 11212. Proposed by David Beckwith, Sag Harbor, NY. Show that for an arbitrary positive integer n Solutions by Christopher Carl Heckman, Arizona State Univeristy, Tempe, AZ
- 11262. Proposed by Ashay Burungale, Satara, Maharashtra, India. In a certain town of population 2n + 1, one knows those to whom one is known. For any set A of n citizens, there is some person amongst the other
- 11343. Proposed by David Beckwith, Sag Harbor, NY. Show that when n is a positive integer, Solutions by Christopher Carl Heckman, Arizona State University, Tempe, AZ
- 11345. Proposed by Roger Cuculi`ere, France. Find all nondecreasing functions f from R to R such that f(x + f(y)) = f(f(x)) + f(y).
- 11350. Proposed by Bhavana Deshpande, Poona College of Arts, Science & Commerce Camp, Pune, India, and M. N. Deshpande, Institute of Science, Nagpur, India. Given a positive integer n and an integer k with
- 11362. Proposed by David Callan, University of Wisconsin, Madison, WI. A bit string arc diagram is an undirected graph in which the vertices are the positions in a single string of bits and the edges are called
- 11424. Proposed by Emeric Deutsch, Polytechnic University, Brooklyn, NY. Find the number of bit strings of length n in which the number of 00 substrings is equal to the number of 11 substrings. For example, when
- 11426. Proposed by M. L. Glasser, Clarkson University, Potsdam, NY. Find (1/14)(9/14)(11/14)
- C. HECKMAN 242 SOLUTIONS TEST 1A
- C. HECKMAN 242 SOLUTIONS TEST 2A
- C. HECKMAN 242 Instructions
- 1.3. Gauss-Jordan Elimination Remember though that we originally wanted to put
- 2.2. Higher-Order Determinants The 1 1 matrix [a] is invertible exactly when a = 0.
- 2.3. Determinants and Elementary Row Operations The bad news is: Row operations can change the
- 2.4. Cramer's Rule If you take system of linear equations
- 4.1. The Vector Space Rn and Subspaces
- 4.2. Linear Combinations and Linear Independence If we know that the vectors v1, v2, . . . , vk are are in
- 4.4. [Null Spaces and] Row and Column Spaces Given a matrix A, what kind of a set is N(A) =
- 6.0. Motivation for Eigenvalues and Eigenvectors We will be looking at calculating eigenvalues and
- 6.2. Diagonalization Recall that, when you have a polynomial in factored
- 5.4. Orthogonal Bases and the Gram-Schmidt Algorithm
- 5.2. Orthogonal Projections and Least Squares Solutions*
- 1.6 and 5.3. Curve Fitting One of the broadest applications of linear algebra is
- C. HECKMAN MAT 271 Solutions Test 2
- 11241. Proposed by Roberto Tauraso, Universit`a di Roma "Tor Vergata", Rome, Italy. Find a closed formula for
- C. HECKMAN MAT 294-I Solutions Test 3
- C. HECKMAN MAT 267 Test 1 SOLUTIONS
- MAT 267 Written Homework #10 SOLUTIONS 13.7 Due: November 17
- MAT 267 Written Homework #9 SOLUTIONS 13.4, 13.5, 13.6 Due: November 10
- C. HECKMAN 267 SOLUTIONS TEST 1F
- C. HECKMAN MAT 267: FALL 2007 SOLUTIONS Test 3
- C. HECKMAN FINAL EXAM: FALL 2007 (1) [15 points] Find the equation of the line which
- C. HECKMAN MAT 267 Solutions Test 2
- MAT 267 Written Homework #3 SOLUTIONS 10.9, 11.1, 11.3, 11.4 Due: September 22
- C. HECKMAN 242 SOLUTIONS TEST 1A
- C. HECKMAN 242 SOLUTIONS TEST 2A
- C. HECKMAN 267 SOLUTIONS QUIZ #1 FORM A
- MAT 267 Written Homework #6 SOLUTIONS 12.2, 12.3, 12.4 Due: October 13
- C. HECKMAN 267 SOLUTIONS TEST 2A
- C. HECKMAN FINAL EXAM: FALL 2007 267 SOLUTIONS
- MAT 267 Written Homework #4 SOLUTIONS 11.4, 11.5, 11.6 Due: September 29
- MAT 267 Written Homework #7 12.5 --12.7 Due: October 27
- MAT 267 Written Homework #7 SOLUTIONS 12.5 --12.7 Due: October 27
- MAT 267 Written Homework #5 SOLUTIONS 11.7, 11.8, 12.1 Due: October 6
- MAT 267 Written Homework #8 SOLUTIONS 13.1 --13.3 Due: November 3
- MAT 267 Written Homework #2 SOLUTIONS 10.5, 10.6, 10.7 Due: September 8
- C. HECKMAN 265 SOLUTIONS TEST 2A