I'm just talking to myself here, but I think condition 1 is "x takes the form (abc)" and condition 2 is "x takes the form (abcde)."
Anybody want to verify?
For any element x \in A_5, we have that [A_5:C_{A_5}(x)]=\begin{cases}
[S_5:C_{S_5}(x)], & \text{condition 1} \\
\frac{1}{2}[S_5:C_{S_5}(x)], & \text{condition 2}
\end{cases}
Basically I want to know what the conditions are.
Note that C is the centralizer.
Homework Statement
Okay so I'm supposed to find the least squares solution of a set of equations, which I can do, but it adds that I must use QR decomposition. I don't really know how to apply QR decomposition to this problem.
Problem: Find the least squares solution of
x_1 + x_2 = 4...
Homework Statement
Let g_1(t) = t - 1 and g_2(t)= t^2+t. Using the inner product on P_2 defined in example 10(b) with t_1=-1,t_2=0,t_3=1, find a basis for the orthogonal complement of Span(g_1, g_2).
Homework Equations
From example 10(b)
\langle p, q \rangle = \sum_{i=1}^{k+1}...
Homework Statement
Let A be an m x n matrix of rank n. Suppose v_1, v_2, ..., v_k \in \mathbb{R}^n and \{v_1, v_2, ..., v_k\} is linearly independent. Prove that \{Av_1, Av_2, ..., Av_k\} is likewise linearly independent.
Homework Equations
The Attempt at a Solution
It says I...
Homework Statement
Prove that C(AB) is a subset of C(A) for matrices A,B, where C denotes column space.
Homework Equations
C(AB) = {b \in \mathbbcode{R}^m: Ax=b is consistent}
The Attempt at a Solution
I don't really know where to start.
Where's the motivation for concluding that there are infinitely many unique infinite cardinal numbers?
I understand the proof for it and accept it, but what other useful implications of this can be drawn in mathematics? It almost seems like Cantor developed this idea in set theory just to say...
I know what you mean, but wouldn't you need an axiom that allows you to "combine" the axioms into one logical statement.
Anywho let me be more specific to dodge your problem then, assume you have only one axiom, the axiom of extensionality from ZFC. Can any truly distinct implications be...
Let (j, k)\in\mathb{N}^2. Without loss of generality, assume j < k.
From here, it's safe to assume that: \forall (j, k), j < \frac{j + k}{2} < k.
Then, p_n < \frac{p_n + p_{n+1}}{2} < p_{n+1}.
Since p_n and p_{n+1} are consecutive primes, \frac{p_n + p_{n+1}}{2} cannot be prime.
Am I missing something, why aren't you using
\zeta(s) = \frac{1}{1-2^{1-s}}\sum_{n\geq1}\frac{(-1)^{n-1}}{n^{s}}
as your definition of zeta?
The definition of zeta in your proof is invalid for \text{Re[s]} < 1.