### Algebra Homework, Edition 2

Let X be a nite set and let R be the ring of functions from X into the eld R of real numbers. Suppose that D is a commutative ring such that D[x] is a principal ideal domain. Let R be a principal ideal domain, and let I and J be ideals of R. IJ denotes the ideal of R generated by the set of all elements of the form ab where a 2I and b Christian Parkinson UCLA Basic Exam Solutions:Linear Solution. Clearly the characteristic polynomial of Ahas a real root since it has odd order. Let be a real root of the characteristic polynomial. Then is an eigenvalue of A. Suppose 0 6=v2R3 is a normalized eivengector corresponding to . Then 2 = 2(v;v) = ( v; v) = (Av;Av) = (v;AtAv) = (v;v) = 1:Thus = 1. If = 1, then we are done.

### Fall 2018 Statistics 201A (Introduction to Probability at

Suppose Cand Cc are the events that the patient has cancer and does not have cancer respectively. Also suppose that + and are the events that the test yields a positive and negative result respectively. By the information given, we have P(j C) = 0:01 P(+jCc) = 0:05 P(C) = 0:02:We need to compute P(Cj+). By Bayes rule, we have P(Cj+) = P(+ \C Homework Two SolutionsIf U is our matrix in question, then because it is real-valued, we have, UT ij = U ji = U = Uy ij; (27) where the rst equality is the de nition of the transpose, the second equality is from the fact that the matrix is real-valued, and the third equality is the de nition of the transpose conjugate. So in IEOR @ IIT BOMBAY1.Suppose a fair coin is tossed repeatedly and heads appeared in the rst 1000 tosses. What is the 4.A real valued function f() on real line R is said to be convex if for any two given reals xand y, denotes the probability that event A j occurs). Then show that must contain at least 2n events.

### J I J J I I - University of South Carolina

J J I I J I Page 1 of 22 Go Back Full Screen Print Close Quit PROBLEMS FROM LINEAR ALGEBRA In the following R denotes the eld of real numbers while C denotes the eld of complex numbers. In general, U,V, and W denote vector spaces. The set of all linear transformations from V into W is denoted by L(V,W), while L(V) denotes the set of Lecture 16 and 17 Application to Evaluation of Real JR = 2(R + R2). The residue computation easily shows that JR = /2. Observe that f(z) = p(z)/q(z), where |p(z)= |z2 1 R2+1,andsimilarly|q(z)= |(z2+1)(z2+4) (R21)(R24). Therefore |f(z) R2 +1 (R2 1)(R2 4) =:MR. This is another lucky break that we Let x be a real number. We denote by [x] the integer Proof. Suppose (a n) is convergent to a:For any >0;there exists N 2N such that ja n aj< =2 whenever n N :Using triangle inequality, we nd that for any n;m N ; ja n a mj ja n aj+ ja m aj< 2 + 2 = :Theorem 0.1. (Completeness of Rn) A sequence of real numbers is convergent if and only if it is a Cauchy sequence. Example 0.4. Prove that the

### Notes on Symmetric Matrices 1 Symmetric Matrices

j max j = max max j j = 0:So max I A 0. 2 In particular, for any symmetric matrix Awe have A kAkI. 1.3 Trace De nition 9 Let Abe an arbitrary d dmatrix (not necessarily symmetric). The trace of A, denoted tr(A), is the sum of the diagonal entries of A. Fact 10 (Linearity of Trace) Let Aand Bbe arbitrary d dmatrices and let ; be scalars. Then Solved:1 => [14 Points) Suppose That A And B Are Real Sym Question:1 => [14 Points) Suppose That A And B Are Real Symmetric N X N Matrices And That (.) Denotes The Usual Inner Product On R. If (Av, V) > 0 For All Nonzero V ER", Prove That There Exist 41,-.. E R And A Basis {0},,Un} Of " Such That If I = 1 If I=j (Avi, Ug) (Bvi, U) - 0 If It I 0 If It J. (Hint:Define A New Inner Product (,y) = (Ax, Y), For All Suppose a, b denote the distinct real roots of the Sep 29, 2020 · Suppose a, b denote the distinct real roots of the quadratic polynomial x 2 + 20x 2020 and suppose c,d denote the distinct complex roots of the quadratic polynomial x 2 20x + 2020. Then the value of ac(a - c) + ad(a - d) + bc(b - c) + bd(b - d) is (A) 0

### group 0, and suppose that u - R is a uniformly

In the case when G is the additive group of real num- bers and I p c, the circle of ideas from the abstract setting denotes the norm of convolution by k on LP(G,) (see [7, Theorem 2.2. Suppose that {kj}j>_l C- Li(G). For each j let T be the convolution operator on LP(G, A) defined by kj. If C is a constant su& thatProof. 2j(p) = p(j)(0) j!. Here p (j) denotes the jth derivative of p, with the understanding that the 0th derivative of p is p. Proof. From Proposition 3.98 we know that the dual basis is a basis of dual space. By de nition of dual basis (3.96), we just need to check if (0.1) j(xk) = (1 (j= k) 0 (j6= k) Note that j(xk) = (x k)(j)(0) j!, hence if j