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(3 + 3 = 6 pts.) Prove or disprove the following statements. If you are proving...
Prove or disprove the following statements. In each case, A and B are both nx n matrices. (a) If C is a 3 x 2 matrix, then C has a left inverse. (b) If Null(AT) = {Õ}, then A is invertible. (c) If A and B are invertible matrices, then A + B is invertible.
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B = C (b) If Bvi,.., Bvh} is a then vi, . ., vk} is a linearly independent set in R". linearly independent set in R* where B is a kx n matrix, 5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B...
3) Prove or Disprove the following statement: If A and B are n x n invertible matrices then A and B are row equivalent. (This is a formal proof problem, be sure to state and justify each step.)
Topology 3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
linear algebra Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
3. Prove the statements that are true and give counterexamples to disprove those that are false. (a). Va,b,n E Z* , if a’ =b}(modn) then a =b(modn). (8 points) (b). If p> 2 and q> 2 are prime, then p? +q must be composite. (12 points)
3. Let X and Y be countably infinite sets. (a) Prove: If X and Y are disjoint then XuY is countably infinite. (b) Is the statement in (a) still true if we remove the hypothesis that X and Y are disjoint? If yes, justify your reasoning with a few sentences. If no, provide a counterexample. (P.S. "Counterexample” means that you have to explain why the example you provide demonstrates that the statement is false.)
Let X, Y, Z be random variables. Prove or disprove the following statements. (That means, you need to either write down a formal proof, or give a counterexample.) (a) If X and Y are (unconditionally) independent, is it true that X and Y are conditionally indepen- dent given Z? (b) If X and Y are conditionally independent given Z, is it true that X and Y are (unconditionally) independent?
3. For each of the following statements decide if it is true or false. If it is true, prove it. If it is false, give an example for which it does not hold. (a) If is an eigenvalue of the (n, n)-matrix A, then 2 - 31+ 512 is an eigenvalue of 21_n - 3A + 5A2 (b) The complex vector V1 = (1 + 1,0,1) is an eigenvector of the matrix [ 2 0 -4 ] A= | 0...
MATLAB: Do the following with the provided .m file (b) Now on the MATLAB prompt, let us create any two 3 × 3 matrices and you can do the following: X=magic(3); Y=magic(3); X*Y matrixMultiplication3by3(X,Y) (c) Now write a new function in MATLAB called matrixMultiplication that can multiply any two n × n matrix. You can safely assume that we will not test your program with matrices that do not have their inner dimensions matched up CODE: function [C] = matrixMultiplicationFor3by3(A,B)...