MAD 3512 Study Guide - Final Guide: Natural Number, Turing Machine, Primitive Recursive Function

Give an example of a 3 times 3-matrix A such that A^2 notequalto 0 but A^3 = 0. (b) Let n N, and suppose that T: R^n rightarrow R^n is a linear transformation such that T^n-1 notequalto 0 but T^n = 0. Show that there exists upsilon R^n such that (upsilon, T upsilon, ..., T^n-1 upsilon) is a basis or R^n. (c) A square matrix B is called nilpotent if B^m = 0 for some m N. Prove that if B is a nilpotent n times n matrix, then B^n = 0.

Find (B + C)A. provided that A = [2 0 5 2 3 -2], B = [2 -2 3 5], C = [0 -2 2 0] Find A^TA if A = [1 5 0 -1 6 -2]. Consider the matrices X = [5 0 5], Y = [5 5 0], Z = [-25 -30 5]. Find scalars a and b such that Z = aX + bY. Use the provided definition to find f(A): If f is the polynomial function, f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n, then for an n times n matrix A, f(A) is defined to be f(A) = a_0I_n + a_1A + a_2A^2 + ... + a_nA^n. Given that f(x) = x^2 - 5x + 3, A = [2 4 0 6]. Find the inverse of the matrix [8 -24 16 -40] Use inverse matrices to solve the system of linear equations {3x - 4y = 22 2x + 3y = -8 Let A = [1 0 -1 2 1 2 -4 2 0], C = [0 0 -1 4 1 2 -4 2 0]. Find an elementary matrix E such that EA=C.

(1 point) In this problem you will calculate | x2 + 5 dx by using the formal definition of the definite integral: ['sor as- en (įrenad f(x) dx = lim n–00 *)Ax|| _k=1 (a) The interval [0, 2] is divided into n equal subintervals of length Ax. What is Ax (in terms of n)? Ax = 2/n (b) The right-hand endpoint of the kth subinterval is denoted x. What is x (in terms of k and n)? x¢ = (0+2k/n (C) Using these choices for x* and Ax, the definition tells us that ('e es deu em (grammar). x2 +5 dx = lim n+00 f(x)dx : What is f(x; )Ax (in terms of k and n)? f(; )ax = (0+2k/ny^2+5)2/m (d) Express Ef(x)Ax in closed form. (Your answer will be in terms of n.) k=1 n Ëso; Jar - ( (e) Finally, complete the problem by taking the limit as n + w of the expression that you found in the previous part. L'ens de (2,maj - ma