Problem 1 (20): Let a, b,c,d ER. Show that (axb) (cx d) = det(A), where A...
Problem 1 (3pts). Let A E Mnn Show that det(A) is the product of the singular values of A Problem 1 (3pts). Let A E Mnn Show that det(A) is the product of the singular values of A
2. (24 pts) Let f(x) = >>= {* Ae Mc 1>C where A,B,C ER, A, B +0. x <C' (a) Show that f is differentiable at x = C. (b) Determine the first four terms of the Taylor series centered at x = C for f(x) using the definition of Taylor series. (c) If possible, find the Taylor series T(2) centered at x = C for f(x). (d) What's the radius and interval of convergence? (e) Find R4(C++). Can you...
Let A. B, C, D є Mnxn(F), and det(A) 0, AC-CA. Prove that A B det ( )) -det(AD CB)
(1) Let (X,d) be a metric space and A, B CX be closed. Prove that A\B and B\A are separated (1) Let (X,d) be a metric space and A, B CX be closed. Prove that A\B and B\A are separated
Гa b c] Let A = d e f . Assume that det(A) = -11, find Igni Ta gol (a) det(2A) (b) det(A-1) c) det(3A-1) (a) det((3A) – 1) (e) detone (a) det(2A) = Click here to enter or edit your answer - 22 (b) det (A-1) = Click here to enter or edit your answer (c) det(3A-1) = Click here to enter or edit your answer (d) det((3A)-1) = Click here to enter or edit your answer a gol...
1. Let A Idef g h i Given that det(A) 1, find det(B) where You should fully justify your answer 3 marks]
1. Let a,b ER with a < b. In this problem we are going to prove that the open interval (a, b) containes infinitely many rational numbers by following these steps (a) First let NEN be an arbitrary rational number. Use the density of the rational numbers to show that (a, b) contains N rational numbers. There is a hint about this in the lecture on the density of rationals.) (b) Now uppose that there are finitely many rational numbers...
Problem 3. Let X be a discrete random variable, gx) - a+ bX+ cX, and let a. b, c be constants. Prove, using the definition of expectation of a function of a random variable, namely , that E(a + bX + cx?) = a + bE(X) + cE(X2)
Problem 1: Let X be a linear space. Let Y CX be a linear subspace. (a) Prove that the map : X+X/Y given by 7(x) = (2) is a linear map. (b) Prove without using the dimension formula or rank-nullity that N = Y. (c) Prove without using the dimension formula or rank-nullity that RX = X/Y.
4. Let A and B be 4 x 4 matrices. Suppose det A= 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(AT)? (d) (4 points) What is det(A-1)? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of AT are linearly independent. [2] and us 6. (6 points) Let vi...