Find all eigenvalues and their corresponding eigenspaces of the following matrix. B=\( \begin{pmatrix}2&-3&1\\ 1&-2&1\\ 1&-3&2\end{pmatrix} \)
In: Advanced Math
Find all eigenvalues and their corresponding eigenspaces of the following matrix. A.\( \begin{pmatrix}3&0\\ 1&2\end{pmatrix} \)
In: Advanced Math
show that for some ring R, the equality a^2−b^2=(a−b)(a+b) holds ∀a,b∈R if and only if R is commutative.
In: Advanced Math
Show that the relation 'a R b if and only if a−b is an even integer defined on the Z of integers is an equivalence relation.
In: Advanced Math
Prove that in a finite cyclic group, each subgroup has size dividing the size of the group. Conversely, given a positive divisor of the size of the group, there is a subgroup of that size.
In: Advanced Math
Let f be measurable and B a Borel set. Then f-1[B] is a measurable set. [Hint: The class of sets for which f-1[E] is measurable is a σ-algebra.
In: Advanced Math
Let G be finite with |G| > 1. If Aut(G) acts transitively on G − {e} then G ∼= (Z/(p))n for some prime p.
In: Advanced Math
How do you calculate the probability of compound events?
In: Advanced Math
Find f(a+h) - f(a) / h where f(x) = 6x - 9
In: Advanced Math
What is the number of group homomorphisms from z12 to z13?
In: Advanced Math
What is the vertex of
y=2x²+6x+4 ? Give details explaination
In: Advanced Math
Given integers a, b, c,
g.c.d.(a, b, c) = 1 if and only if g.c.d.(a, b) = 1 and g.c.d.(a, c) = 1
In: Advanced Math
The number of involutions in G is |G|/4, and every right coset of a Sylow 2-subgroup S of G not contained in NG(S) contains exactly one involution.
In: Advanced Math
An integer n is called even if n = 2m for some integer m, and odd if n + 1 is even. Prove the following statements:
(a) An integer cannot be both even and odd.
(b) Every integer is either even or odd.
(c) The sum or product of even integers is an even integer. What can you say about the sum or product of odd integers?
In: Advanced Math