Some mathematical notation used in Math 375 Set notation Examples: cfw_x : x R, x > 4, or also cfw_x R : x > 4, denotes the set of all real numbers which are greater than 4. x E means the element x belongs to the set E. x A means the element x does not be
Mathematics 375 Some problems about vectors 1. If a quadrilateral OABC in R2 is a parallelogram having A and C as opposite vertices, prove that A + 1 (C A) = 1 B . What geometric theorem about 2 2 parallelograms follows from this equation? Show that every
Mathematics 375 Topics in Linear Algebra and Multivariable Calculus Fall 2005 Course Outline: I. Linear spaces (ch. 1): Review of vector algebra , Vector spaces and subspaces, dependent and linearly independent sets, bases and dimension, inner products an
Diagonalizable matrices Theorem 1. Let A be a real n n matrix and suppose that there exists a basis of Rn which consists of eigenvectors; i.e. there are n linearly independent vectors u1 ,., un , and real numbers 1 , ., n so that (1) Aui = i ui , i = 1, .
Math 375 Review problems III Do the following problems from the book. 3.6: 4a, c, 6. 3.11: 5. 3.17: 1c, 2c, 3c. 4.4: 2, 5. 4.8: 4, 5, 6, 8, 11. Review the handout on determinants to do No. 9 in 4.8. or come up with your own proof. 5.11: No 1, 5, 7, 8, 12.
Math 375 Review problems II 1. Let V be the vector space of all vectors x R4 with the property x1 + x2 + x3 + x4 = 0. Then u1 = (1, 0, 0, 1) and u2 = (0, 1, 1, 0) are two linearly independent vectors in V . (i) Find an orthonormal basis of the span of u1
Mathematics 375 Homework problems from the textbook 1.5: 9, 10, 11, 12, 13, 14, 23-28. Extra Credit: 1.5 No. 29 and 30. 1.10: 1-10. Make a sketch for each of the sets in No. 1-10. 1.10: 13, 19, 20, 22, 23 (a,e,f,g,h), 24 1.13: 4, 5, 11, 16. 1.17: 2, 3, 8.
Math 375 Midterm test III 1. Let 0 0 2 -1 A = 0 0 1 2 2 0 0 5 . 7 4 5 3
(i) Evaluate the determinant of A. (ii) Find the determinants of AT , A-1 (if A is invertible), A100 , (AT )100 . 2. Let T : Rn Rn be a linear transformation. Prove: The number 0 is a
Math 375 Midterm Exam II October 27, 2005 Note: There are five problems. Do as much as you can during this exam. Take this exam sheet home. If you do not have time to finish all the problems you should complete the exam at home and hand in what you did fo
Math 375 Midterm Exam I In class part C1. Prove by induction that for all n = 1, 2, 3, . . .
n
k(k + 1) =
k=1
n(n + 1)(n + 2) . 3
n(n+1)(n+2) .) 3
(That is, 1 2 + 2 3 + 3 4 + + n(n + 1) =
C2. Let S = cfw_v1 , . . . , vn . (i) Write down the correct defini
MATHEMATICAL INDUCTION
Suppose we want to prove a mathematical statement A(n) which is formulated for every natural number n N, i.e. n=1,2,3,4,. One very often uses the following method (referred to as "mathematical induction", or simply "induction") to v
MORE ON DETERMINANTS
Math 375
Preliminaries on permutations Let Zn = cfw_1, 2, 3, . . . , n. A function : Zn Zn which is bijective (i.e. one-to-one and onto1 ) is called a permutation of the numbers 1, 2, 3, . . . , n (or a permutation on n "letters"). We