MthSc 853 (Linear Algebra) Dr. Macauley HW 1 Due Monday, January 24th, 2011 (1) (a) Show that there are no proper subfields of Q. (b) Show that Q( 2) = cfw_a + b 2 | a, b Q is a field. (2) Let X be a vector space over a field K. Let 0 be the zero element
MTHSC 853, Linear Algebra Spring Semester 2011 MWF 12:201:10 Martin E005 General information
Instructor: Email: Web: Phone: Office: Office hours: Text: Dr. Matthew Macauley macaule@clemson.edu http:/www.math.clemson.edu/~macaule/ (864) 656-1838 Martin O-3
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MthSc 853 (Linear Algebra) Dr. Macauley HW 11 Due Wednesday, April 20th, 2011 Throughout, X is a finite-dimensional Euclidean space. (1) Define the index of a real symmetric matrix A to be the number of strictly positive eigenvalues minus the number of st
MthSc 853 (Linear Algebra) Dr. Macauley HW 2 Due Wednesday, February 2nd, 2011 (1) Let S be a set of vectors in a finite-dimensional vector space X. Show that S is a basis of X if every vector of X can be written in one and only one way as a linear combin
MthSc 853 (Linear Algebra) Dr. Macauley HW 3 Due Monday, February 7th, 2011 (1) Let T : X U be a linear map. Prove the following: (a) The image of a subspace of X is a subspace of U . (b) The inverse image of a subspace of U is a subspace of X. (2) Prove
MthSc 853 (Linear Algebra) Dr. Macauley HW 4 Due Monday, February 14th, 2011 (1) Show that whenever meaningful, (i) (ST ) = T S (ii) (T + R) = T + R (iii) (T -1 ) = (T )-1 . Here, S denotes the transpose of S. Carefully describe what you mean by "whenever
MthSc 853 (Linear Algebra) Dr. Macauley HW 5 Due Monday, February 21st, 2011 (1) Let T : X U , with dim X = n and dim U = m. Show that there exist bases B for X and B for U such that the matrix of T in block form is Ik 0 M= , 0 0 where Ik is the k k ident
MthSc 853 (Linear Algebra) Dr. Macauley HW 6 Due Wednesday, March 2nd, 2011 (1) Let Sn denote the set of all permutations of cfw_1, . . . , n. (a) Prove that if Sn is a transposition, then sgn( ) = -1. (b) Let Sn , and suppose that = k 1 = 1 , where i , j
MthSc 853 (Linear Algebra) Dr. Macauley HW 7 Due Wednesday, March 9th, 2011 (1) Prove the following properties of the trace function: (a) tr AB = tr BA for all m n matrices A and n m matrices B. (b) tr AAT = a2 for all n n matrices A. ij (2) Find the eige
MthSc 853 (Linear Algebra) Dr. Macauley HW 8 Due Wednesday, March 16th, 2011 (1) Consider the following matrices: A= -2 0 0 -2 , B= 0 -2 -2 0 , C= 3 -1 1 1 .
(a) Determine the characteristic and minimal polynomials of A, B, and C. (b) Determine the eigenv
MthSc 853 (Linear Algebra) Dr. Macauley HW 9 Due Friday, April 1st, 2011 (1) Consider the following matrix: Mn = 0 In-1 -a0 -an-1 where an = a1 a2 . . . an (a) Show that the characteristic polynomial of Mn is PMn (t) = tn + an-1 tn-1 + + a1 t + a0 . Here,
MthSc 853 (Linear Algebra) Dr. Macauley HW 10 Due Monday, April 11th, 2011 (1) Let X be a finite-dimensional real Euclidean space. We say that a sequence cfw_An of linear maps converges to a limit A if limn |An - A| = 0. (a) Show that cfw_An converges t