2. A = Z, the set of integers with usual addition and multiplication. Prove if true...
(6 pts) Alternate construction of the integers from the natural numbers. Suppose that the natural numbers N = {0,1,2, ...} ations. We do not yet have a notion of subtraction or the cancellation law for addition (if x+y = x+ z, then y = 2) and for multiplication given with the usual addition and multiplication oper negative numbers, though we do have are Define a relation R on N2 as follows (a, b) R (c, d) if and only if...
8.) Consider the integers Z. Dene the relation on Z by x y if and only if 7j(y + 6x). Prove: a.) The relation is an equivalence relation. b.) Find the equivalence class of 0 and prove that it is a subgroup of Z with the usual addition operator on the integers. 8.) Consider the integers Z. Define the relation ~ on Z by x ~ y if and only if 7)(y + 6x). Prove: a.) The relation is an...
(1) Consider Z with the addition and multiplication mod 3 as usual. Let R=ZgxZg. Define (a, b)+ (a',b) (aab+and(a,b)((aa-bab +a'b) (a) Show that (R, +) is a commutative ring. b) Show that (1,0) is the identity element for the multiplication. c) Show that the equation 22 hs exactly two solutions in R Bonus Problem) Show that (R, +,.) is a field. (Hint: To find multiplicative inverse, first show that a2 + b2メ0 if (a, b)メ(0.0). Then compute (a, b).(a,-b).) (1)...
linear algebra 1. Determine whether the given set, along with the specified operations of addition and scalar multiplication, is a vector space (over R). If it is not, list all of the axioms that fail to hold. a The set of all vectors in R2 of the form , with the usual vector addition and scalar multiplication b) R2 with the usual scalar multiplication but addition defined by 31+21 y1 y2 c) The set of all positive real numbers, with...
Ax=O Unique solution (trivial solution x-0) No free variables Infinitely many (nontrivial) solutions Some free variables Every column of A is pivot column | (=> rank(A) = # of columns of A Some columns of A are not pivot columns rank(A)< #of columns of A You can use the above figure to answer the following questions are about homogeneous systems Ax-0. Answer TRUE or FALSE. If the answer is FALSE, choose FALSE with the appropriate counterexample, i.e example that shows...
Question 1 (10 Marks) This question consists of 10 true false ansers. In cach ease, answer true if the statement is always true and false otherise. If a statement is false, 1. The set rER0 isa group under the binary operation o defined ad-be is a group under matrix addition. 3. Tho sot eRzs not an Abelian group under the binary erplain why. There is no need to show working for true statements. by a ob vab. 2. The set...
Show that the following are not vector spaces: (a) The set of all vectors [x, y] in R^2 with x ≥ y, with the usual vector addition and scalar multiplication. ------------------------------------------------[a b] (b) The set of all 2×2 matrices of the form [c d] in where ad = 0, with the usual matrix addition and scalar multiplication. I need help with this question. Could you please show your work and the solution.
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
Write a formal proof to prove the following conjecture to be true or false. If the statement is true, write a formal proof of it. If the statement is false, provide a counterexample and a slightly modified statement that is true and write a formal proof of your new statement. Conjecture: 15. (12 pts) Let h: R + RxR be the function given by h(x) = (x²,6x + 1) (a) Determine if h is an injection. If yes, prove it....
IT a) If one row in an echelon form for an augmented matrix is [o 0 5 o 0 b) A vector bis a linear combination of the columns of a matrix A if and only if the c) The solution set of Ai-b is the set of all vectors of the formu +vh d) The columns of a matrix A are linearly independent if the equation A 0has If A and Bare invertible nxn matrices then A- B-'is the...