4) If A and B are mxn matrices and k and t are real numbers, prove...
3) Let u33 2, and wE-ls -1] 2 yes, compute them. If any of them is not defined, explain why not. b) Treating u, v', and w' as matrices, are the products v'u, and ne detined? If yes, compute them. If any of them is not defined, explain why not. 4) If A and B are mxn matrices and k and t are real numbers, prove that a) (A +B)k- Ak+ BAk b) A(k +t) Ak+ At Note that to...
it veetors délfhed above 2) Find the length of vector i-2 3 s 3) Let u--3 v 3 2l, and w [5 -1l 2 a) Treating u, v', and w, as vectors, are the inner products uw, v,u, and 1.w, defined? If yes, compute them. If any of them is not defined, explain why not. b) Treating u,v, and w' as matrices, are the products n', p'u and zm defined? If yes, compute them. If any of them is not...
Problem 1 Let {ak} and {bk} be sequences of positive real numbers. Assume that lim “k = 0. k+oo bk 1. Prove that if ) bk converges, so does 'ak k=1 k=1 2. If ) bk diverges, is it necessary that ) ak diverges? k=1 k=1
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
please answer 2a(i) only 2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 × 9 matrices. The (i, j)-entry of the matrix B is given by i *j - 1. The (i,j)-entry of the matrix A equals 0 if i +j is divisible by and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u- [9,...
1) The sequence ak is defined as ao = 4, a1 = 5, Ak+1 = 30k – 20k-1,k=1,2,... what is the general formula for ax? 2) The sequence bk is defined as bo = a, b1 = ß, bk+1 = 4bk – 4bk-1, what is the general formula for bk? Hint: Prove the corresponding matrix is similar to [ ] To compute k 2.1 you need to use the following fact: Pk+1 = 2pk +2k == Pk = (P. +...
3. (10%) Let C = AB, where A and B are both n by n matrices. The element located at row i and column of C is represented by C, and computed as C, = A, B, + A,B2, + ... + 4,B, a) Express C, using the X (summation) notation. b) Evaluate c, if Ak = 2 and B, = 3 for all k, k = 1,2,...n.
ssume A and B are invertible nxn matrices and k is a scalar. Prove the following. a.) If A is invertible, then 14-1 (1/(|4). (AB),I=1시1
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that 4 #d) Use the inequality in #b) to prove the inequality which is the triangle inequality #2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in...
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...