Homework Answers
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn...
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn = 0. Show that lim(anon) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is...
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If...
Exercise 2.3.7: Let {xn} and {yn} be bounded sequences. a) Show that {Xn+yn} is bounded. b)...
Exercise 2.3.7: Let {xn} and {yn} be bounded sequences. a) Show that {Xn+yn} is bounded. b) Show that (lim inf xn) + (lim inf yn) < lim inf (Xn tyn). noo Hint: Find a subsequence {Xn; +yn;} of {Xn +yn} that converges. Then find a subsequence {Xnm;} of {Xn;} that converges. Then apply what you know about limits. n->00 c) Find an explicit {{n} and {yn} such that noo (lim inf xn) + (lim inf yn) <lim inf (Xn+yn). noo...
please solve 42.7: (a),(b),(c) 42.7. Consider = N()(a - bN(t – (m)] (a) Show that small...
please solve 42.7: (a),(b),(c)
42.7. Consider = N()(a - bN(t – (m)] (a) Show that small displacements from equilibrium satisfy the following linear delay-differential equation: dN (1) = -aN (6- m. dt (b) Ifm > 0, we will not find the general solution. Instead, let us look for special solutions of the form of exponentials, N (1) = e. Show that r = -ae-rim 184 Population Dynamics-Mathematical Ecology (c) Show graphically there are two real solutions if the delay is...
a) Show that [a,b] | ab. b) Let d be a common divisor of a and...
a) Show that [a,b] | ab.
b) Let d be a common divisor of a and b. Show that
.
c) Prove that (a,b)*[a,b] = ab.
d) Prove that if c is a common multiple of a and b, then
such that k[a,b] = c.
e) Suppose that c is a common multiple of a and b. Show that ab
| (a,b)*c
Defn: Let m e Z. We say that m is a common multiple of a and b if...
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0 4. Let A and B be n x n such that B 1-A and A2 = A. Show that AB-BA-0 4. Let A and B be n x n such that B = 1-A and...
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0 4. Let A and B be n x n such that B 1-A and A2 = A. Show that AB-BA-0
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0
4. Let A and B be n x n such that B...
P.2.16 Let V= span {AB-BA : A, B E Mn. (a) Show that the function tr...
P.2.16 Let V= span {AB-BA : A, B E Mn. (a) Show that the function tr : M,,-> C is a linear transformation. (b) Use the dimension theorem to prove that dim ker tr = n2-1. (c) Prove that dim V = n2-1. (d) Let Eij=eie), every entry of which is zero except for a 1 in the (i, j) position. Show that k,-OikEil for l i, j, k, n. (e) Find a basis for V. Hint: Work out the...
Can you solve No.6 6. Let (a.)and b) be bounded sequences in R .a. Prove that lima. +İimb, siim (a, + b.) s ima, + İim br b. Prove that lim (-a)lima . Given an example to show that equality need n...
Can you solve No.6
6. Let (a.)and b) be bounded sequences in R .a. Prove that lima. +İimb, siim (a, + b.) s ima, + İim br b. Prove that lim (-a)lima . Given an example to show that equality need not hold in (a) If o, and b, are positive for all n, prove that lim (a)s(im a)mb). provided the product on the right is 7. not of the form 0 oo. b. Need equality hold in (a)?
6....
4. Let A and B be 4 x 4 matrices. Suppose det A= 4 and det(AB)...
4. Let A and B be 4 x 4 matrices. Suppose det A= 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(AT)? (d) (4 points) What is det(A-1)? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of AT are linearly independent. [2] and us 6. (6 points) Let vi...
ANSWER 5,6 & 7 please. Show work for my understanding and upvote. THANK YOU!! Problem 5....
ANSWER 5,6 & 7 please. Show work for my understanding and
upvote. THANK YOU!!
Problem 5. (3 pts) Let {x,n} be a bounded sequence of real numbers and let E = {xn : n E N}. Prove that lim inf,,0 In and lim inf, Yn are both in E. Hint: Use the sequential characterization of the closure, i.e., Proposition 3.2 from class. Problem 6. (3 pts) As usual let Q denote the set of all rational numbers. Prove that R....
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn = 0. Show that lim(anon) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If...
Exercise 2.3.7: Let {xn} and {yn} be bounded sequences. a) Show that {Xn+yn} is bounded. b) Show that (lim inf xn) + (lim inf yn) < lim inf (Xn tyn). noo Hint: Find a subsequence {Xn; +yn;} of {Xn +yn} that converges. Then find a subsequence {Xnm;} of {Xn;} that converges. Then apply what you know about limits. n->00 c) Find an explicit {{n} and {yn} such that noo (lim inf xn) + (lim inf yn) <lim inf (Xn+yn). noo...
please solve 42.7: (a),(b),(c)
42.7. Consider = N()(a - bN(t – (m)] (a) Show that small displacements from equilibrium satisfy the following linear delay-differential equation: dN (1) = -aN (6- m. dt (b) Ifm > 0, we will not find the general solution. Instead, let us look for special solutions of the form of exponentials, N (1) = e. Show that r = -ae-rim 184 Population Dynamics-Mathematical Ecology (c) Show graphically there are two real solutions if the delay is...
a) Show that [a,b] | ab.
b) Let d be a common divisor of a and b. Show that
.
c) Prove that (a,b)*[a,b] = ab.
d) Prove that if c is a common multiple of a and b, then
such that k[a,b] = c.
e) Suppose that c is a common multiple of a and b. Show that ab
| (a,b)*c
Defn: Let m e Z. We say that m is a common multiple of a and b if...
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0 4. Let A and B be n x n such that B 1-A and A2 = A. Show that AB-BA-0
4. Let A and B be n x n such that B = 1-A and A2 = A. Show that AB BA = 0
4. Let A and B be n x n such that B...
P.2.16 Let V= span {AB-BA : A, B E Mn. (a) Show that the function tr : M,,-> C is a linear transformation. (b) Use the dimension theorem to prove that dim ker tr = n2-1. (c) Prove that dim V = n2-1. (d) Let Eij=eie), every entry of which is zero except for a 1 in the (i, j) position. Show that k,-OikEil for l i, j, k, n. (e) Find a basis for V. Hint: Work out the...
Can you solve No.6
6. Let (a.)and b) be bounded sequences in R .a. Prove that lima. +İimb, siim (a, + b.) s ima, + İim br b. Prove that lim (-a)lima . Given an example to show that equality need not hold in (a) If o, and b, are positive for all n, prove that lim (a)s(im a)mb). provided the product on the right is 7. not of the form 0 oo. b. Need equality hold in (a)?
6....
4. Let A and B be 4 x 4 matrices. Suppose det A= 4 and det(AB) = 20. (a) (4 points) What is det B? (b) (4 points) Is B invertible? Why or why not? (c) (4 points) What is det(AT)? (d) (4 points) What is det(A-1)? 5. (6 points) Let A be an n x n invertible matrix. Use complete sentences to explain why the columns of AT are linearly independent. [2] and us 6. (6 points) Let vi...
ANSWER 5,6 & 7 please. Show work for my understanding and
upvote. THANK YOU!!
Problem 5. (3 pts) Let {x,n} be a bounded sequence of real numbers and let E = {xn : n E N}. Prove that lim inf,,0 In and lim inf, Yn are both in E. Hint: Use the sequential characterization of the closure, i.e., Proposition 3.2 from class. Problem 6. (3 pts) As usual let Q denote the set of all rational numbers. Prove that R....
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