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Let R be a commutative ring with unity and r ∈ R . Prove that R[x] / (x-r) ≅ Rusing the Fundamental Theorem of Homomorphism.
Hence show that R[x,y]/(y-r)≅ R[x].
Hence show that R[x,y]/(y-r)≅ R[x].
Let D = {f (x,y) + g(x,y)i | f,g ∈Z [x,y] }⊆ C [x,y] . Check whether D is a UFD
or not.
or not.
A lawyer commutes daily from his suburban home to his midtown office . on the average the trip one way takes 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed.
a) what is probability that a trip will take atleast 1/2 hour?
b) If the office opens at 9:00 a.m. and he leaves his house at 8:45 a.m. daily what percentage of the time is he late for work?
c) if he leaves the house at 8:35 a.m and coffee is served at the office from 8:50 a.m until 9:00 a.m. , what is the probability that he misses coffee?
d) find the length of time above which we find the slowest 15% of the trips.
e) find the probability that 2 of the next 3 trips will take atleast 1/2 hour?
a) what is probability that a trip will take atleast 1/2 hour?
b) If the office opens at 9:00 a.m. and he leaves his house at 8:45 a.m. daily what percentage of the time is he late for work?
c) if he leaves the house at 8:35 a.m and coffee is served at the office from 8:50 a.m until 9:00 a.m. , what is the probability that he misses coffee?
d) find the length of time above which we find the slowest 15% of the trips.
e) find the probability that 2 of the next 3 trips will take atleast 1/2 hour?
Using Xo = 0 find an approximation to one of the zeros of x^3 − 4x +1 = 0 by using Birge Vieta Method. Perform two iterations
Express the complex number in trigonometric form.
-3 + 3 square root of threei
-3 + 3 square root of threei
Let V be the vector space of 2× 2 matrices over R. Check whether the subsets
W1 = { (a,1) ,(0,-a) | a ∈ R} and W2 = { (a,-a), (0,b)| a,b∈ R}
are subspaces over R. For those sets which are subspaces, find their dimension and a
basis over R.
W1 = { (a,1) ,(0,-a) | a ∈ R} and W2 = { (a,-a), (0,b)| a,b∈ R}
are subspaces over R. For those sets which are subspaces, find their dimension and a
basis over R.
Let V be a vector space over a field F and let T : V → V be a linear operator. Show
T(W) ⊂ W for any subspace W of V if and only if there is a λ ∈ F such that Tv = λv
for all v ∈ V .
T(W) ⊂ W for any subspace W of V if and only if there is a λ ∈ F such that Tv = λv
for all v ∈ V .
Let R = Z[ √2 ] and M = {a + b √2 ∈ R | 5|a and 5|b}
i) Show that M is an ideal of R .
ii) Show that if 5|a or 5|b , then (a^2 + b^2), for a,b∈Z
iii) Hence show that if N is an ideal of R properly containing M , then N = R .
iv) Show that R/M is a field, and give two distinct non-zero elements of this
field.
i) Show that M is an ideal of R .
ii) Show that if 5|a or 5|b , then (a^2 + b^2), for a,b∈Z
iii) Hence show that if N is an ideal of R properly containing M , then N = R .
iv) Show that R/M is a field, and give two distinct non-zero elements of this
field.
Define a relation R on Z , by R = {( n, n + 3 k) | k ∈Z} .
Check whether R is an equivalence relation or not. If it is, find all the distinct
equivalence classes. If R is not an equivalence relation, define an equivalence
relation on Z .
Check whether R is an equivalence relation or not. If it is, find all the distinct
equivalence classes. If R is not an equivalence relation, define an equivalence
relation on Z .
Math
Find the inverse of the matrix {(3, 1, 2),(-2,3,-5),(1,2,4)} using Gauss Jordan Method
