Let A. B, C, D є Mnxn(F), and det(A) 0, AC-CA. Prove that A B det...
Problem 5. Let a < b and c > 0 and let f be integrable on [ca, cb]. Show that f c Ca where g(a) f(ex)
4. Assume that A, B E Mnxn(R). Prove or disprove each of the following statements. (a) If AB is a product of elementary matrices, then A is a product of elementary matrices. (b) If R is the RREF of A, then det A = det R. (c) If det A-det B, then A = B.
Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A
Let R(A,B,C,D) be a relation with FDs F = {A—B, AC, C-A, B,C, ABC-D} Which of the following statements is correct ? (2 points) Select one: G = {A-B, B-C, C-A, AC=D } is a canonical cover of F H = { AC, CA, BC,BD} is a canonical cover of F. o F is a canonical cover of itself. O G and H are canonical covers of F. None of the above.
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact. (b) Prove that for any є > 0 there exists some N > 0 so that for any x E A we have (c) Prove that A is totally bounded. (d) Prove that A is compact (2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact....
Гa b c] Let A = d e f . Assume that det(A) = -11, find Igni Ta gol (a) det(2A) (b) det(A-1) c) det(3A-1) (a) det((3A) – 1) (e) detone (a) det(2A) = Click here to enter or edit your answer - 22 (b) det (A-1) = Click here to enter or edit your answer (c) det(3A-1) = Click here to enter or edit your answer (d) det((3A)-1) = Click here to enter or edit your answer a gol...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
True or False det(AB) det(BA) det(A B) det(A) + det(B) det(CA) c det(A) = C det((AB)T) det(A) det(B) det(B) => A = B det(A) det(A) det(A) A triangular matrix is nonsingular if and only if its diagonal entries are all nonzero.
Problem 2. Let C[0, 1] be the set of all continuous functions from [0, 1] to R. For any f, g є Cl0, 11 define - max f(x) - g(z) and di(f,g)-If(x) - g(x)d. a) Prove that for any n 2 1, one can find n points in C[O, 1 such that, in daup metric, the distance between any two points is equai to 1. b) Can one find 100 points in C[0, 1] such that, in di metric, the...