Prove/disprove that for any linear function f there is only one matrix [A] for which f(x) = [A]x for all x.
Prove/disprove that for any linear function f there is only one matrix [A] for which f(x) = [A]x ...
Consider the function ?:ℤ×ℤ×ℤ→ℤ, defined by ?(?,?,?)=?2?−?3. a) Is ?f a one-to-one function? Prove or disprove. b) Is ?f an onto function? Prove or disprove.
please help in detail 1. Prove or disprove the following statements: a. For any matrix A € Rmxn with Rank(A) = r, A and AT have the same set of singular values. b. For any matrix A ER"X", the set of singular values is the set of eigenvalues.
2. Prove that, for any linear function f(x) of a random variable x, no matter what its p.d.f.p(x), it must be true that (f(x)) = f((x)). (2 marks) Note: Saying f(x) is linear means f(x) = ax + b where a and b are constants.
(1) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) Compute the following determinant (1) (4 pts) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) (2+2 pts) Compute the following determinant (123) (100 A= 1023 B=020 003 co c
Prove or disprove: there is a one-to-one map from A to B if and only if there is a onto map from B to A.
(-8,00) defined as f(x)= x + 6x +1. Prove or disprove that it is 1-1 11. We are given a function S:(-3,00) and/or onto, using algebraic evidence.
1- Prove or disprove. (X,Y are topological spaces, A, B are subsets of a topological space X, Ā denotes the closure of the set A, A' denotes the set of limit points of the set A, A° denotes the interior of the set A, A denotes the boundary of the set A.) (a) (AUB) = A'U Bº. (b) f-1(C') = (F-1(C))' for any continuous function f :X + Y and for all C CY. (c) If A° ), then A°=Ā.
1. Prove that the function f: X → Y is injective if and only if it satisfies the following condition: For any set T and functions g: T → X and h : T → X, o g = f o h implies g = h.
2. Suppose that A is an rn x n matrix and b є С". Prove that the linear system CSA, b) is consistent if and only if r(A) = r(Ab) 2. Suppose that A is an rn x n matrix and b є С". Prove that the linear system CSA, b) is consistent if and only if r(A) = r(Ab)
linear algebra// only the top question 1. Find the Fourier series of f(x) = x + 2 over the interval (0,25). To receive full credit, you must show all work when integrating. 3. (12 points) Prove each of the following parts. a) Prove that the characteristic equation of a 2 x 2 matrix A can be expressed as