Linear operator Definition and 110 Threads

Homework Statement Suppose that T:W -> W is a linear transformation such that Tm+1 = 0 but Tm ≠ 0. Suppose that {w1, ... , wp} is basis for Tm(W) and Tm(uk) = wk, for 1 ≤ k ≤ p. Prove that {Ti(uk) : 0 ≤ i ≤ m, 1 ≤ j ≤ p} is a linearly independent set.Homework Equations The Attempt at a Solution...

Homework Statement Let x=(x1,x2,x3), Ax:=(x2-x3,x1,x1+x3), Bx:=(x2,2x3,x1) Find: (3B+2A2)x. Homework Equations The Attempt at a Solution Warning: I have no idea what I'm doing! (3B+2A2)x = 3Bx+2A2x 3Bx = (3x2,6x3,3x1) Now to find 2A2x. Considering that an index has a...

Given a vector v\in H: R2->R3 which is a function from R2 to R3, and an operator G: H->H on the space H in which v lives, what is the most commonly used symbol to refer to this mapping, i.e., G(v)=G\otimes v or something else?

Homework Statement Let T: \ell^{2} \rightarrow \ell be defined by T(x)=x_{1},\frac{1}{2}x_{2},\frac{1}{3}x_{3},\frac{1}{4}x_{4},...} Show that the range of T is not closed The Attempt at a Solution I figure that I need to find some sequence of x_{n} \rightarrow x such that...

I don't know how to start this problem. Since it's a bi-implication, I need to show each statement implies the other. I started playing around with the definitions of inner product and norm directly, but it's not going anywhere...

Homework Statement Let T be a linear operator on a finite dimensional vector space V. Suppose ||T(x)|| = ||x|| for all x in V, prove that T is one to one. Homework Equations ||T(x)||^2 = <T(x),T(x)> ||x||^2 = <x,x> The Attempt at a Solution Suppose T(x) = T(y) x, y in V Then...

Homework Statement Verify that any square matrix is a linear operator when considered as a linear transformation. Homework Equations The Attempt at a Solution If a square matrix A\inℂ^{n,n} is a linear operator on the vector space C^{n}, where n ≥ 1, then the square matrix A is...

Homework Statement Let L be a linear operator such that: L[1, 1, 1, 1] = [2, 1, 0, 0] L[1, 1, 1, 0] = [0, 2, 1, 0] L[1, 1, 0, 1] = [1, 2, 0, 0] L[1, 0, 1, 1] = [2, 1, 0, 1] a) Find [L] b) Compute L[1, 2, 3, 4] Homework Equations The Attempt at a Solution I used another...

I'm having difficulty understanding the concepts presented in the following question. I'm given a matrix, [2,4,1,2,6; 1,2,1,0,1; ,-1,-2,-2,3,6; 1,2,-1,5,12], which is the matrix representation of a linear operator from R5 to R4. The question asks me to find a basis of the image and the...

T is a linear operator from the space of 2 by 2 matrices over the complex plane to the complex plane, that is T: mat(2x2,C)\rightarrowC, given by T[a b; c d] = a + d T operates on a 2 by 2 matrix with elements a, b, c, d, in case that isn't entirely clear. So T gives the trace of the...

Let P belongs to Mnn be a nonsingular matrix and Let L:Mnn>>>Mnn be given by L(A) = P^-1AP for all A in Mnn. Prove that L is an invertible linear operator. I have no clues how to start this question. What do I need to prove for this question? and why

Homework Statement Let L(x) a linear operator defined by setting the diagonal elements of x to zero. What will be the representation of this operator to the following basis set? x E X. X denote the set of all real symmetric 3x3 matrices. Homework Equations L*y=x L=x*inv(y)...

Hi, If T is a bounded linear operator on a Hilbert space, what can we say about the spectra of T and T* (\sigma(T)=\{\lambda:T-\lambda I is not invertible})?

Homework Statement Much as the title says, I need to construct an example of a linear operator on Hilbert space with empty spectrum. I can very easily construct an example with empty point-spectrum (e.g. the right-shift operator on l_2), but this has very far from empty spectrum. If I...

I just want to test/verify my knowledge of change of basis in a linear operator.. (it's not a homework question). Suppose I have linear operator mapping R^2 into R^2, and expressed in the canonical basis (1,0), (0,1). Suppose (for the sake of discussion) that the linear operator is given by...

Technically this isn't homework, but just something I saw another user state without proof in a very different thread. I believe, however, that it is specific enough to pass as a "homework question" so I thought I'd pretend that it was and post it here, because I'm getting a bit frustrated with...

Homework Statement Let T be a linear operator on an inner product space V. Prove that ||T(x)|| = ||x|| for all xεV iff <T(x),T(y)> = <x,y> for all x,yεV Homework Equations The Attempt at a Solution <T(x),T(y)> = <x,y> so <x,T^*T(y)> =<x,y> This seems too simple. What...

Homework Statement Show that any linear operator \hat{O} can be decomposed as \hat{O}=\hat{O}'+i\hat{O}'', where \hat{O}' and \hat{O}'' are Hermitian operators. Homework Equations Operator is Hermitian if: T=T^{\dagger}The Attempt at a Solution I don't know where to start :\ Should I try...

Homework Statement Let L be the operator on P_3(x) defined by L(p(x)) = xp'(x)+p"(x) if p(x) = a_0(x)+a_1(x)+a_2(1+x^2) calculate L^n(p(x)) Homework Equations stuck between 2 possible solutions i) as powers of x decrease the derivatives of p(x) increase ii) as derivatives...

I have been re-reading some linear analysis the last week, and I have been playing around a bit with the linear subspace of \mathbb{R}^\mathbb{R} that you get with the basis {sin x, cos x}. It didn't take me long to realize that the family of transformations parametrized by theta T_\theta ...

If L is the following first-order linear differential operator L = p(x) d/dx then determine the adjoint operator L* such that integral from (a to b) [uL*(v) − vL(u)] dx = B(x) |from b to a| What is B(x)? sorry.. on my book there's only self-adjointness i don't quiet understand what is...

Homework Statement T is a linear operator on a finite dimensional vector space. then N(T*T)=N(T). the null space are equal. Homework Equations The Attempt at a Solution this is my method, but its does not work if dim(R(T))=0. I'm only concerned with showing N(T*T) \subseteq...

I have a question about the invertibility of a linear operator T. In Friedberg's book, Theorem 6.18 (c) claims that if B is an orthonormal basis for a finite-dimensional inner product space V, then T(B) is an orthonromal basis for V. I don't understand the proof, I think the book only...

Hello, If f is a morphism between two vector spaces, we say it is linear if we have: 1) f(x+y) = f(x) + f(y) 2) f(ax) = af(x) Now, if f is the convolution operator \ast , we have a binary operation, because convolution is only defined between two functions. Can we still talk about linearity in...

I would be grateful for some help/tips/with this question. Let (V,<,>) be a complex inner product space with an orthonormal basis {v1,v2,...vn}. Let L:V------>V be a linear operator. Explain what is meant by saying that L is self-adjoint. Let A=a_ij be the matrix representing L with respect...

Homework Statement Let V be a finite-dimensional vector space over the field F and let S and T be linear operators on V. We ask: When do there exist ordered bases a and b for V such that [S]a = [T]b? Prove that such bases exist if and only if there is an invertible linear operator U on V...

Homework Statement Given a linear operator T, show that if rank(T^2)=rank(T), then the range and null space are disjoint. So I know that I can form a the same basis for range(T^2) and range(T), but I'm not sure where to go from there.

It seems to me that http://en.wikipedia.org/wiki/Schur_decomposition" relies on the fact that every linear operator must have at least one eigenvalue...but how do we know this is true? I have yet to find a linear operator without eigenvalues, so I believe every linear operator does have at...

Homework Statement Show that the range \mathcal{R}(T) of a bounded linear operator T: X \rightarrow Y is not necessarily closed. Hint: Use the linear bounded operator T: l^{\infty} \rightarrow l^{\infty} defined by (\eta_{j}) = T x, \eta_{j} = \xi_{j}/j, x = (\xi_{j}). Homework Equations...

Definitions: A linear operator T on a finite-dimensional vector space V is called diagonalizable if there is an ordered basis B for V such that [T]_B is a diagonal matrix. A square matrix A is called diagonalizable if L_A is diagonalizable. We want to determine when a linear operator T on a...

Homework Statement For each of the following inner product spaces V (over F) and linear transformation g:=V \rightarrow F, find a vector y such that g(x) = <x,y> for all x element of V. The particular case I'm having trouble with is: V=P2(R), with <f,h>=\int_0^{1} f(t)h(t)dt ...

Homework Statement Let V be an inner product space and T:V->V a linear operator. Prove that if T is normal, then T and T* have the same kernel (T* is the adjoint of T). Homework Equations The Attempt at a Solution Let us assume x is in the kernel of T. Then, TT*x =T*Tx = T*0= 0...

Let T is a linear transformation from a vector space V to V itself. The dimension of V, denoted by dim(V), is finite. If the rank of T, denoted by rk(T), is equivalent to the rank of TT, i.e., rk(T)=rk(TT) why is the intersection of image of T(denoted by im(T)) and the kernel of T(denoted...

prove that a linear operator.. T(f):=\frac{\mathrm{d^2f} }{\mathrm{d} x^2}+2\frac{\mathrm{df} }{\mathrm{d} x} T(kf)=kT(f) part: T(kf):=k\frac{\mathrm{d^2f} }{\mathrm{d} x^2}+2k\frac{\mathrm{df} }{\mathrm{d} x}=k(\frac{\mathrm{d^2f} }{\mathrm{d} x^2}+2\frac{\mathrm{df} }{\mathrm{d}...

Homework Statement Show that if an operator A satisfies A2 - A + I = 0 then A has an inverse. Express A-1 as a simple polynomial of A. Homework Equations I'm not sure that this is relevant, but A-1=1/(detA)TrC where TrC is the transpose of the matrix of cofactors. Also: If detA = 0 then the...

Statement Let S be a linear operator S: U-> U on a finite dimensional vector space U. Prove that Ker(S) = Ker(S^2) if and only if Im(S) = Im(S^2) So, I'm really not sure about how to prove this properly. I have a few ideas, but this one seemed to make sens intuitively to me. So, I'm...

Butkov's book present the theory of linear operators this way: Suppose a linear operator \alpha transforms a basis vector \hat{\ e_i} into some vector \hat{\ a_i}.That is we have \alpha\hat{\ e_i}=\hat{\ a_i}......(A) Now the vectors \hat{\ a_i} can be represented by its co-ordinates...

[b]1. Let T be the linear operator on R^n that has the given matrix A relative to the standard basis. Find the spectral decomposition of T. A= 7, 3, 3, 2 0, 1, 2,-4 -8,-4,-5,0 2, 1, 2, 3 [b]3. eigen values are 1 (mulitplicity 1), -1 (mult. 1), 3 (mult. 2). And associated eigen...

Homework Statement Let V be a vector space of dimension n. And the linear operators E=A^0, A^1, A^2, ... A^(n-1) are linearly independent. Prove that there exists a v in V such that V=<v, Av, A^2v, ..., A^(n-1)v> Homework Equations The Attempt at a Solution Here are something that...

Homework Statement Let V be a finite vector space, and A, B be any two linear operator. Prove that, rank A = rank B + dim(Im A \cap Ker B) The Attempt at a Solution Since rank A = dim I am A dim(Im B)+ dim(Ker B)=dim V dim(Im A + Ker B)=dim(Im A)+dim(Ker B)-dim(Im A \cap Ker B) It...

Homework Statement http://img389.imageshack.us/img389/9272/33055553mf5.png The Attempt at a Solution Via induction: for n=1 equality holds now assume that Vn=Jn. I introduce a dummy variable b and the fundamental theorem of calculus and change order of integration: V_{n+1}f(t)...

Greetings, can someone check if I'm doing this correctly? I have to find the standard matrix for the linear operator T defined by the formula. For example, T(x1,x2,x3) = (x1 + 2x2 + x3, x1+ 5x2, x3) Is the matrix I want just simply, T = 1 2 1 1 5 0 0 0 1 I'm basing this...

Homework Statement Suppose T \in L(\textbf{C}^3) defined by T(z_{1}, z_{2}, z_{3}) = (z_{2}, z_{3}, 0). Prove that T has no square root. More precisely, prove that there does not exist S \in L(\textbf{C}^3) such that S^{2} = T. Homework EquationsThe Attempt at a Solution I showed in a...

In finite dimensions, a matrix can be decomposed into the sum of rank-1 matrices. This got me thinking - in what situations can a bounded linear operator mapping between infinite dimensional spaces be written as an (infinite) sum of rank-1 operators? eg, let A be a bounded linear operator...

Homework Statement 1. Let T be a linear operator on an inner product space V. Let U = TT*. Prove that U = TU*. 2. For a linear operator T on an inner product space V, prove that T*T = T0implies T = T0. Homework Equations The Attempt at a Solution 1. This appeared at the...

Homework Statement This is from a linear algebra textbook I'm reading. I don't know whether there is universal agreement as to how notation should read for this, so this thread may well be meaningless. But I'll just post it to see if anyone can decipher what the question means. I'm asked to...

Homework Statement Show that the range of the linear operator defined by the equations is not all of R3, and find a vector that is not in the range Homework Equations w1 = x - 2y + z w2 = 5x - y + 3z w3 = 4x + y + 2z The Attempt at a Solution can I just show that it does not have...

Hi, I am looking for eigen functions of the linear operator L defined by L=(-2i(\nablaf).\nabla -i\nabla^2f +(\nablaf)^2) and here f is an abitary function of x,y,z

[SOLVED] linear algebra determinant of linear operator Homework Statement Let T be a linear operator on a finite-dimensional vector space V. Define the determinant of T as: det(T)=det([T][SIZE="1"]β) where β is any ordered basis for V. Prove that for any scalar λ and any ordered basis...

Let T:V -> V be a linear operator on a finite-dimensional inner product space V. Prove that rank(T) = rank(T*). So far I've proven that rank (T*T) = rank(T) by showing that ker(T*T) = ker(T). But I can't think of how to go from there.