Serge Ballif
MATH 535 Homework 1
August 31, 2007
1) (a) Find integers x0 and y0 such that 4x + 11y = 1. (b) Explain how all pairs (x, y) of integers 4x + 11y = 1 are related to the pair you found. (a) The ordered pair (x0 , y0 ) = (3, -1) satisfie
Algebra 2 A
Assignment U1L01A1
10points
1. If x = 11.4 and y = -8.2 what is 2x + 3y?
2(11.4) + 3(-8.2)
-1.8
2. Explain which computation you would do first in the problem
13 + 5[2(1025 -19)] -711
-The first thing that you would do would be 10/2. Since you
Polynomial Functions and the Quaternions Serge Ballif Prerequisites and Definitions Polynomial Functions
Polynomial Functions and the Quaternions
Serge Ballif ballif@math.psu.edu
The Pennsylvania State University
January 18, 2008
Outline
Polynomia
Polynomials and Quaternions
The main reference for the talk is A First Course in Noncommutative Rings by T. Y. Lam. Definition. A ring R is a set together with two binary operations "+" and " such that the following three properties hold. 1. (R, +) f
Polynomials and Quaternions Serge Ballif, January 24, 2008
The main reference for the talk is A First Course in Noncommutative Rings by T. Y. Lam.
1
1.1
Prerequisites and Definitions
Rings
Rings Definition 1. A ring R is a set together with two bi
Mutually Orthogonal25,Latin Squares Serge Ballif, February 2008
The main reference for the talk is Discrete Mathematics Using Latin Squares by Charles F. Laywine and Gary L. Mullen
1
1.1
3 Challenges
Challenge I
Challenge I Consider the 16 aces, k
Mutually Orthogonal Latin Squares Serge Ballif 3 Challenges Definitions
Mutually Orthogonal Latin Squares
Serge Ballif ballif@math.psu.edu
The Pennsylvania State University
MOLS
February 26, 2008
3 Challenges Challenge I Challenge II Challenge II
Serge Ballif
MATH 535 Homework 12
December 7, 2007
1) Let M, N be free R-modules with bases u1 , . . . , um and v1 , . . . , vn respectively. Let the transformations S : M M , T : N N , have matrices A = (aij ) and B = (bkl ) with respect to the
Serge Ballif
MATH 535 Homework 11
November 30, 2007
1-2) Let I be an ideal of the ring R, and let M, N be R-modules such that IN = 0. Then show that, as R/I modules, M R N M/IM R/I N with the isomorphism "given by" m R n [m] R/I n. Remark: Be sur
Serge Ballif
MATH 535 Homework 10
November 9, 2007
1. Let A () denote the characteristic polynomial of A Mat(F). Show that A is diagonalizable over F if and only if (a) A () splits over F into a product of linear factors and (b) whenever (A - 0 I
Serge Ballif
MATH 535 Homework 8
October 26, 2007
The goal of this homework set is to establish the, perhaps surprising, analogue for finite dimensional vector spaces for the problems of last weeks homework. For this set, we fix a linear transform
Serge Ballif
MATH 535 Homework 9
November 2, 2007
1. Show that if V is a vector space with minimal polynomial P (X) and v V has order P (X), then there is a subspace U of V such that V = F[X]v U , and the minimal polynomial of U divides P (X). H
Serge Ballif
MATH 535 Homework 7
October 19, 2007
1. This exercise shows (part of) the fact every ring R can be considered as a subring of a ring with identity element. We mimic putting the subring i into the full ring of Gaussian integers: On S :
Serge Ballif
1. Define the linear transformation U on j n := exp(2ij)/n), i = 0, . . . , n - 1. a) Show that the sum
MATH 535 Homework 6
October 12, 2007
Cn by v F v, where nF is the Vandermonde matrix for the values
n j (n )k = k=1
Sj =
i n
n
Serge Ballif
MATH 535 Homework 5
September 28, 2007
1. a) Let A = (aij ) Matnm (F), such that for each i, n aij = 1. Prove that 1 is an eigenj=1 value of A. b) Show that, if instead n aij = 1 for j = 1, . . . , n, then 1 is still an eigenvalue of
Serge Ballif
MATH 535 Homework 4
September 21, 2007
1. (Schanuel) Use the fact that a polynomial P (X) F[x] has at most deg P roots in F to show that the Vandermonde determinant is non-zero when the entries are distinct. Recall that
1 c1 . . 1 c
Serge Ballif
MATH 535 Homework 3
September 14, 2007
1. a) Use what you know about linear transformations to prove that dim Col(AB) dim Col(A). b) Use what you know about linear transformations to prove that dim Col(AB) dim Col(B). a) Let fA and
Serge Ballif
MATH 535 Homework 2
September 7, 2007
Let the m n matrix A reduce to the matrix R in reduced row-echelon form. Let R have columns Cj1 , . . . , Cjs corresponding to the independent variables Xj1 , . . . , Xjs of the linear system of
Daniel Akech
Linear Algebra: Math 535
Homework 2
09/17/2014
Problem. 1 Prove that if two vectors are linearly dependent, one of them is a scalar
multiple of the other.
Let u and v be two vectors in a vector space V. If they are linearly dependent, there a