Linear operator Definition and 117 Threads

I am trying to figure out what the matrix of this linear operator would be: T:M →AMB where A, M, B are all 2X2 matrices with respect to the standard bases of a 2x2 matrix viz. e11, e12 e21 and e22. Any ideas? Il know it should be 2X8 matrix. I am trying to teach myself Abstract algebra using...

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 [e_{j}:j\in N] be an orthonormal set in a Hilbert space H and \lambda_{k} \in R with \lambda_{k} \rightarrow 0 Then show that Ax=\sum_{j=1}^{\infty} \lambda_{j}(x,e_{j})e_{j} Defines a compact self adjoint operator H \rightarrow H The Attempt at a Solution...

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 an inner product space V and W be a T-invariant subspace of V. If W is both T and T* invariant, prove that (T_{W})* = (T*)_{W}. Note that T_{W} denotes the restriction of T to W Homework Equations \forallx\inW, T_{W}(x) = T(x)...

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 T be a linear operator on a vector space V,and let W be a T-invariant subspace of V.Define \ddot{T}:V⁄W→V⁄W by \ddot{T}(v+W)=T(v)+W for any v+W∈V⁄W. Prove that if both TW and \ddot{T} are diagonisable and have no common eigenvalues, then T is diagonisable. Homework...

Homework Statement (5.5)V = Z_{\xi_{1}} \oplus Z_{\xi_{2}} \oplus ... \oplus Z_{\xi_{k}}, the basis for V is: \xi_{1}, \eta(\xi_{1}), ..., \eta(\xi_{1})^{p_{1} - 1} \xi_{2}, \eta(\xi_{2}), ..., \eta(\xi_{2})^{p_{2} - 1} . . . \xi_{k}, \eta(\xi_{k}), ..., \eta_(\xi_{k})^{p_{k} - 1}...

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 ...

Homework Statement Let T, linear operator on non fin. dim. Vector Space over field of complex numbers such that there exists T*, adjoint of T. If <T(x),x> = 0 for all x in V then T=T_0 T_0 s.t. T_0(x)=0 for all x in V 2. The attempt at a solution Consider <T(x''),x''> s.t. x''=x' + y...

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...

Homework Statement Let 1 \leq p \leq \infty and let (X,\Omega, \mu) be a \sigma-finite measure space. For \phi \in L^\infty(\mu) , define M_\phi on L^p(\mu) by M_\phi f = \phi f \forall f \in L^(X,\Omega,\mu). I need to find the following: \sigma(M_\phi) , \sigma_ap(M_\phi), and...

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...