Homework Statement
Show that x^{n}+y^{n}=z^{n} has a nontrivial solution if and only if the equation \frac{1}{x^{n}}+\frac{1}{y^{n}}=\frac{1}{z^{n}} has a nontrivial solution.
Homework Equations
By nontrivial solutions, it is implied that they are integer solutions.
The Attempt at a Solution...
Homework Statement
Let V be a finite dimensional vector space and T is an operator on V. Assume μ_{T}(x) is an irreducible polynomial. Prove that every non-zero vector in V is a maximal vector.
Homework Equations
μ_{T}(x) is the minimal polynomial on V with respect to T.
The Attempt at...
Homework Statement
This question came out of a section on Correspondence and Isomorphism Theorems
Let V be a vector space and U \neq V, \left\{ \vec{0} \right\} be a subspace of V. Assume T \in L(V,V) satisfies the following:
a) T(\vec{u} ) = \vec{u} for all \vec{u} \in U
b) T(\vec{v} + U) =...
Homework Statement
Let V be a vector space over the field F and consider F to be a vector space over F in dimension one. Let f \in L(V,F), f \neq \vec{0}_{V\rightarrow F}. Prove that V/Ker(f) is isomorphic to F as a vector space.
Homework Equations
L(V,F) is the set of all linear maps...
I hope I'm posting this in the right place.
Homework Statement
Let V be a finite dimensional vector space over a field F and T an operator on V. Prove that Range(T^{2}) = Range(T) if and only if Ker(T^{2}) = Ker(T)
Homework Equations
Rank and Nullity theorem:
dim(V) = rank(T) +...
Homework Statement
Determine the equilibrium temperature distribution for a one-dimensional rod composed of two different materials in perfect thermal contact at x=1. For 0<x<1, there is one material (cp=1, K0=1) with a constant source (Q=1), whereas for the other 1<x<2 there are no sources...