Let
be an arbitrary function and A
X.
i) Show that A
ii) Give an example to show that in general A =
.
iii) Show that, if
is injective, then A =
iv) Show that, if X and Y are modules;
is a homomorphism of modules and A is a submodule of X such that
ker,
then we also have A =
Let be an arbitrary function and A X. i) Show that A ii) Give an example...
Q: Let L be a finite-dimensional Lie algebra over C with universal enveloping algebra U(L), and let V and W be L-modules. (1) Define what is meant by an L-module homomorphism o: V the modules V and W to be isomorphic W and explain what it means for (ii) Explain what is meant by a submodule S of V and describe the factor module V/S. V W be an L-module homomorphism Let (iii) Show that ker(ø) is a submodule of...
Being F the subset of of the hemi-symmetric matrices ( such as ). i) Show that F is a subspace of . ii) Determine the dimension of F. iii) Determine the base of F. iv) Being the application that corresponds to each matrix of F the vector of . Determine the matrix that represents T regarding the base of the previous question (iii) and the canonical base of . v) Determine if T is injective. vi) Determine if T is surjective....
Let be a prime and let be the set of rational numbers whose denominator (when written in lowest terms) is not divisible by . i) Show, with the usual operations of addition and multiplication, that is a subring of . ii) Show that is a subring of . iii) Is a field? Explain. iv) What is where is the set of all fractions with denominator a power of We were unable to transcribe this imageWe were unable to transcribe this...
Let be i.i.d. . Define the sample mean and the sample variance by and . (i) Find the distribution of and for i = 1, ... , n. (ii) Show that and are independent for i = 1, ... , n. (iii) Hence, or otherwise, show that and are independent. 7l N (μ, σ2) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Show that we have the analogous bound for the case of an arbitrary, but countable, number of events [Hint: use the limit properties of the probability function.] We were unable to transcribe this imageWe were unable to transcribe this image
Let be a field of characteristic and in . i.) Suppose has a zero in . Show splits in and find the factorization of ii.)Suppose does not have a zero in . Let be a zero of in an extension of . Show splits in and find a factorization of . We were unable to transcribe this imageWe were unable to transcribe this imagef(x) = XP- We were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let be an arbitrary mapping satisfying the properties (S1) - (S4) of Theorem (at the end). Beyond that let Show that the following statements apply to all u, v ∈ Rn. The Theorem: For the scalar product, vectors u, v, w∈ Rn : We were unable to transcribe this imageWe were unable to transcribe this imageu_u We were unable to transcribe this image u_u
Let X ~ Poisson(). Show that as , converges in distribution to a random variable Y and find the distribution of Y. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Give examples, if possible, of the following. i) A set with a supremum but no maximum. ii) A decreasing sequence so that does not exist iii) An increasing sequence so that does not exist. We were unable to transcribe this image(an) n=1 We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let , and let be a polynomial. Show that if is an eigenvalue of , then is an eigenvalue of . Hint: this follows from the more precise statement that if is a non-zero eigenvector for for the eigenvalue , then is also an eigenvector for for the eigenvalue . Prove this. TEL(V) PEPF) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...