Recent content by toni07

A n×n matrix A is decomposable if there exists a nonempty proper subset I⊆{1,2,...,n} such that aij=0 whenever i∈I and j∉I. I only know the definition of maximal vector which is: A vector z such that the minimal polynomial of the operator T with respect to z = the minimal polynomial of the...

Homework Statement Let V be a finite-dimensional vector space over F, and let T : V -> V be a linear operator. Prove that T is indecomposable if and only if there is a unique maximal T-invariant proper subspace of V. Homework Equations The Attempt at a Solution I tried using the...

Re: Prove that every nonzero vector in $V$ is a maximal vector for $T$ The question in the title: Prove that every nonzero vector in $V$ is a maximal vector for $T$.

Let $T: V \rightarrow V$ be a linear operator on a finite-dimensional vector space $V$ over $F$. Assume that $_{\mu T}(x) \in F[x]$ is an irreducible polynomial. I don't understand how assuming that the minimal polynomial is prime helps to prove the question. Please help.

Homework Statement Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6): (0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ) a) Find the T-cyclic subspace generated by each standard basis vector...

Re: Assume that the field F has at least n distinct elements $a_1, …, a_n$ Sorry, I didn't realize I omitted the last part. b) Let $b_1,...,b_n$ in F be arbitrary (not necessarily distinct). Prove that there exists a unique polynomial g(x) of degree ≤ n - 1 in x with coefficients in F such...

Let n be a positive integer, and for each $j = 1,..., n$ define the polynomial $f_j(x)$ by f_j(x) = $\prod_{i=1,i \ne j}^n(x-a_i)$ The factor $x−a_j$ is omitted, so $f_j$ has degree n-1 a) Prove that the set $f_1(x),...,f_n(x)$ is a basis of the vector space of all polynomials of degree ≤ n -...

a) V = U_1 ⊕ U_-1 where U_λ = {v in V | T(v) = λv} b) if V = M_nn(R) and T(A) = A^t then what are U_1 and U_-1 When V is a vector space over R, and T : V -> V is a linear operator for which T^2 = IV .

My main problem is finding a(x), and b(x), and I still don't know how.

I already used the Euclidean algorithm to get the gcd which is 2, but I need help with the rest of the question.

a) Over the field Q, compute h(x) = gcd(f(x), g(x)), and find polynomials a(x) and b(x) such that h(x) = a(x)f(x) + b(x)g(x). (b) Same question over the field F_2 = {0, 1}. I already computed the gcd(f(x), g(x)) to be 2, but I don't really understand how I'm supposed to find a(x) and b(x)...

Let V and W be two finite-dimensional vector spaces over the field F. Let B be a basis of V, and let C be a basis of W. For any v 2 V write [v]B for the coordinate vector of v with respect to B, and similarly [w]C for w in W. Let T : V -> W be a linear map, and write [T]C B for the matrix...

Prove that G is a subspace of V ⊕ V and the quotient space (V ⊕ V) / G is isomorphic to V. Let $V$ be a vector space over $\Bbb{F}$, and let $T : V \rightarrow V$ be a linear operator on $V$. Let $G$ be the subset of $V \oplus V$ consisting of all ordered pairs $(x, T(x))$ for $x$ in $V$. I...

Let A be an m x n matrix with entries in R. Let T_A : R^n -> R^m be the linear map T_A(X) = A_X. Let U be the solution set of the homogeneous linear system A_X = O. Let W be the set of all vectors Y such that Y = A_X for some X in R^n. I don't really know what I'm supposed to do here, any help...

Assume also that S + T = Iv and that S ∘ T = Ov = T ∘ S. Prove that V = X ⊕ Y where X = range(S) and Y = range(T). I don't understand how to go about it, please help.