Homework Answers
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7. If Cs(r and C6=(y), then show that: x to a homomorphism. So, r...
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show...
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show that φ is an onto homomorphism and find the kernel of φ (10 Points)
Consider d on R defined by d(x, y) = ?|x − y|. (1) Show that (R,...
Consider d on R defined by
d(x, y) = ?|x − y|.
(1) Show that (R, d) is a metric space.
(2) Show that the path γ(t) = t, t ∈ [0, 1] has infinite
length.
Remark: On (2), you only need to verify by the partitions of
equal distances. Although this is slightly different from the
actual definition, it indeed implies that length equals to
infinity, by using some techniques in the Riemann sum (e.g.
refining a partition). This...
#4 please, thank you! 3. Let f : [0, 1] → R be uniformly continuous, so...
#4 please, thank you!
3. Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that |x – y <DE =\f(x) – f(y)] < e for every x, y € [0, 1]. The graph of f is the set Gf = {(x, f(x)) : x € [0, 1]}. Show that Gf has measure zero (9 points). 4. Let f : [0, 1] x [0, 1] → R be...
please answer both!! thank you 6. Is the transformation T: R → R defined T(x, y)...
please answer both!! thank you
6. Is the transformation T: R → R defined T(x, y) = (x + y, x - y + 1) a linear transformation? [3 marks] 8. Let A = 5 6 Find the eigenvalues and ONE of the corresponding 21 eigenvectors of A. [5 marks]
Thank you to any who can help. - Let f(x, y) = sin r cos y...
Thank you to any who can help.
- Let f(x, y) = sin r cos y +2.14 and let C be the path in the ry plane that follows the arc of y = sin x from 6, 1) to (7,0). Then (a) Find the gradient of f, that is, find F=Vf. (b) Explain why Green's Theorem cannot be applied to find | F. dr Jc (c) Use a different method to find F. dr
7. Let S = [0, 1] × [0, 1] and f : S → R be defined by f(x, y) = ( x + y, if x 2 ≤ y ≤ 2x 2 , 0, elsewhere. Show that f is integrable over S and calculate R S f(z)dz.
7. Let S = [0, 1] × [0, 1] and f : S → R be defined by f(x, y) = ( x + y, if x 2 ≤ y ≤ 2x 2 , 0, elsewhere. Show that f is integrable over S and calculate R S f(z)dz.
Please explain and how to do it ? Thank you so much!! Problem 4. If X...
Please explain and how to do it ? Thank you so much!!
Problem 4. If X is an exponential random variable with parameter = 2, compute the probability density function of the random variable Y defined by Y = log X +1.
Please solve from a) to e), thank you. 1. Let R be a com ive ring...
Please solve from a) to e), thank you.
1. Let R be a com ive ring of charact a) Prove that (x+y)P-y. [3] b) Deduce that the map фр: R R, фр(x)-x", is a ring homomorphism. [1] c) Compute Op in the case R is the ring Zp. [2] d) Prove that φp is injective if R has no zero-divisors. [2] e) Give an example of a commutative ring of characteristic p such that фр is not surjective. [3]
Please help! Thank you so much!!! 1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-mo...
Please help! Thank you so much!!!
1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-module P → A such that the following diagram commutes (ie, g。h homomorphism h: (a) Suppose that P is a projective R-module. Show that every short exact sequence 0 → ABP -0 is split exact (and hence B A P). (b) Prove...
Please show work. Thank you!! 6. Simplify: ya va 7. Solve for 2: 32+1 = 2722-7...
Please show work. Thank you!!
6. Simplify: ya va 7. Solve for 2: 32+1 = 2722-7 8. If in x = -4, then x = ... 9. If log x = -2, then x = ... 10. 2 log r - 3 log y =
10. Define φ : R2 → R by φ(x,y-x + y for (x,y) E R2. Show that φ is an onto homomorphism and find the kernel of φ (10 Points)
Consider d on R defined by
d(x, y) = ?|x − y|.
(1) Show that (R, d) is a metric space.
(2) Show that the path γ(t) = t, t ∈ [0, 1] has infinite
length.
Remark: On (2), you only need to verify by the partitions of
equal distances. Although this is slightly different from the
actual definition, it indeed implies that length equals to
infinity, by using some techniques in the Riemann sum (e.g.
refining a partition). This...
#4 please, thank you!
3. Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that |x – y <DE =\f(x) – f(y)] < e for every x, y € [0, 1]. The graph of f is the set Gf = {(x, f(x)) : x € [0, 1]}. Show that Gf has measure zero (9 points). 4. Let f : [0, 1] x [0, 1] → R be...
please answer both!! thank you
6. Is the transformation T: R → R defined T(x, y) = (x + y, x - y + 1) a linear transformation? [3 marks] 8. Let A = 5 6 Find the eigenvalues and ONE of the corresponding 21 eigenvectors of A. [5 marks]
Thank you to any who can help.
- Let f(x, y) = sin r cos y +2.14 and let C be the path in the ry plane that follows the arc of y = sin x from 6, 1) to (7,0). Then (a) Find the gradient of f, that is, find F=Vf. (b) Explain why Green's Theorem cannot be applied to find | F. dr Jc (c) Use a different method to find F. dr
Please explain and how to do it ? Thank you so much!!
Problem 4. If X is an exponential random variable with parameter = 2, compute the probability density function of the random variable Y defined by Y = log X +1.
Please solve from a) to e), thank you.
1. Let R be a com ive ring of charact a) Prove that (x+y)P-y. [3] b) Deduce that the map фр: R R, фр(x)-x", is a ring homomorphism. [1] c) Compute Op in the case R is the ring Zp. [2] d) Prove that φp is injective if R has no zero-divisors. [2] e) Give an example of a commutative ring of characteristic p such that фр is not surjective. [3]
Please help! Thank you so much!!!
1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-module P → A such that the following diagram commutes (ie, g。h homomorphism h: (a) Suppose that P is a projective R-module. Show that every short exact sequence 0 → ABP -0 is split exact (and hence B A P). (b) Prove...
Please show work. Thank you!!
6. Simplify: ya va 7. Solve for 2: 32+1 = 2722-7 8. If in x = -4, then x = ... 9. If log x = -2, then x = ... 10. 2 log r - 3 log y =
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