Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J.
A=\( \begin{pmatrix}1&-1&-1\\ 0&0&-1\\ 0&1&2\end{pmatrix}\: \)
In: Advanced Math
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