What is Linear operator: Definition and 117 Discussions

In mathematics, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping



{\displaystyle V\to W}
between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.
If a linear map is a bijection then it is called a linear isomorphism. In the case where


{\displaystyle V=W}
, a linear map is called a (linear) endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that


{\displaystyle V}


{\displaystyle W}
are real vector spaces (not necessarily with


{\displaystyle V=W}
), or it can be used to emphasize that


{\displaystyle V}
is a function space, which is a common convention in functional analysis. Sometimes the term linear function has the same meaning as linear map, while in analysis it does not.
A linear map from V to W always maps the origin of V to the origin of W. Moreover, it maps linear subspaces in V onto linear subspaces in W (possibly of a lower dimension); for example, it maps a plane through the origin in V to either a plane through the origin in W, a line through the origin in W, or just the origin in W. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.
In the language of category theory, linear maps are the morphisms of vector spaces.

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  1. Z

    Do these two statements imply an underlying induction proof?

    Here is one proof $$\forall u\in U\implies Tu\in U\subset V\implies T^2u\in U\implies \forall m\in\mathbb{N}, T^m\in U\tag{1}$$ Is the statement above actually a proof that ##\forall m\in\mathbb{N}, T^m\in U## or is it just shorthand for "this can be proved by induction"? In other words, for...
  2. Euge

    POTW A Linear Operator with Trace Condition

    Let ##V## be a finite dimensional vector space over a field ##F##. If ##L## is a linear operator on ##V## such that the trace of ##L\circ T## is zero for all linear operators ##T## on ##V##, show that ##L = 0##.
  3. J

    Linear operator in 2x2 complex vector space

    Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2? ____________________________________________________________ An ordered basis for C2x2 is: I don't...
  4. U

    I Orthogonality of Eigenvectors of Linear Operator and its Adjoint

    Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...
  5. F

    I What conditions are needed to raise a linear operator to some power?

    Each operator has a domain, so for a power of an operator to exist, the domain of the operator must remain invariant under the operation. Is that correct? mentor note: edited for future clarity
  6. L

    A Is the Frechet Derivative of a Bounded Linear Operator Always the Same Operator?

    I understand the Frechet derivative of a bounded linear operator is a bounded linear operator if the Frechet derivative exists, but is the result always the same exact linear operator you started with? Or, is it just "a" bounded linear operator that may or may not be known in the most general case?
  7. Leonardo Machado

    I Chebyshev Differentiation Matrix

    Hi everyone. I am studying Chebyshev Polynomials to solve some differential equations. I found in the literature that if you have a function being expanded in Chebyshev polynomials such as $$ u(x)=\sum_n a_n T_n(x), $$ then you can also expand its derivatives as $$ \frac{d^q u}{dx^q}=\sum_n...
  8. M

    A Understanding the Spectrum of a Linear Operator

    Hi PF! What is meant by the spectrum of a linear operator ##A##? I read somewhere that if ##0## belongs in the spectrum, then ##A## is not invertible. Can anyone finesse this for me? I read the wikipedia page, but this was tough for me to understand. Perhaps illustrating with a simple example...
  9. S

    I Historical basis for: measurement <-> linear operator?

    What is the history of the concept that a measurement process is associated with a linear opeartor? Did it come from something in classical physics? Taking the expected value of a random variable is a linear operator - is that part of the story?
  10. Adgorn

    Proof regarding the image and kernel of a normal operator

    Homework Statement Show that if T is normal, then T and T* have the same kernel and the same image. Homework Equations N/A The Attempt at a Solution At first I tried proving that Ker T ⊆ Ker T* and Ker T* ⊆ Ker, thus proving Ker T = Ker T* and doing the same thing with I am T, but could not...
  11. KennethK

    Calculate the spectrum of a linear operator

    <mod note: moved to homework> Calculate the spectrum of the linear operator ##T## on ##B(\ell^1)##. $$T(x_1,x_2,x_3,\dots)=(\sum_{n=2}^\infty x_n, x_1, x_2, x_3, \dots)$$I think the way to do it is to find the point spectrums of ##T## and its adjoint ##T^*##. But I don't know how to calculate...
  12. Adgorn

    Linear algebra problem: linear operators and direct sums

    Homework Statement Homework Equations N/A The Attempt at a Solution I proved the first part of the question (first quote) and got stuck in the second (second quote). I defined Im(E1) as U and Im(E2) as W and proved that v=u+w where v ∈ V, u ∈ U and w ∈ W. After that however I got stuck at...
  13. A

    I Prove the sequence is exact: 0 → ker(f) → V → im(f) → 0

    Problem: Let f ∶ V → V be a linear operator on a finite-dimensional vector space V . Prove that the sequence 0 → ker(f) → V → im(f) → 0 is exact at each term. Attempt: If I call: a: 0 → ker(f), b: ker(f) → V, c: V → im(f), d: im(f) → 0. Then the sequence is exact at: ker(f) if...
  14. W

    Linear Differential Equations and Linear Operator Problem

    Homework Statement I'm not sure how to approach this. The question involves linear operators and a non-homogenous differential equation. Here is the question: https://s15.postimg.org/cdmw80157/Capture.png Homework Equations They are given in the question The Attempt at a Solution I really...
  15. M

    A What Is the Spectrum of a Linear Operator in L2 Spaces?

    http://<img src="https://latex.codecogs.com/gif.latex?L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left&space;|t-s&space;\right&space;|f(s)ds" title="L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left |t-s \right |f(s)ds" />[/PLAIN] I have many doubts on linear operator. How I can find a spectrum of a linear...
  16. S

    I Proving the Linearity of the Curl Operator in Electromagnetic Theory

    Hi, I stumbled upon thinking that "Is curl operator a linear operator" ? I was reading EM Theory and studied that the electromagnetic field satisfies the curl relations of E and B. But if the operator was not linear then how can a non linear operator give rise to a linear solution. Thus it...
  17. G

    MHB Proving $(T^2-I)(T-3I) = 0$ for Linear Operator $T$

    Problem: Let $T$ be the linear operator on $\mathbb{R}^3$ defined by $$T(x_1, x_2, x_3)= (3x_1, x_1-x_2, 2x_1+x_2+x_3)$$ Is $T$ invertible? If so, find a rule for $T^{-1}$ like the one which defines $T$. Prove that $(T^2-I)(T-3I) = 0.$ Attempt: $(T|I)=\left[\begin{array}{ccc|ccc} 3 &...
  18. G

    Prove that linear operator is invertible

    Homework Statement Let \mathcal{A}: \mathbb{R^3}\rightarrow \mathbb{R^3} is a linear operator defined as \mathcal{A}(x_1,x_2,x_3)=(x_1+x_2-x_3, x_2+7x_3, -x_3) Prove that \mathcal{A} is invertible and find matrix of \mathcal{A},A^{-1} in terms of canonical basis of \mathbb{R^3}. Homework...
  19. S

    Unitary and linear operator in quantum mechanics

    Given a transformation ##U## such that ##|\psi'>=U|\psi>##, the invariance ##<\psi'|\psi'>=<\psi|\psi>## of the scalar product under the transformation ##U## means that ##U## is either linear and unitary, or antilinear and antiunitary. How do I prove this? ##<\psi'|\psi'>## ##= <U\psi|U\psi>##...
  20. G

    How Does the Linear Operator \(\phi\) Transform Matrices to Polynomials?

    Homework Statement Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where B= \begin{bmatrix} 3 & -2 \\ 2 & -2 \\ \end{bmatrix} Find rank,defect and one basis of an image and kernel of linear operator...
  21. S

    Interval of convergence of a linear operator

    Homework Statement A function of a hermitian operator H can be written as f(H)=Σ (H)n with n=0 to n=∞. When is (1-H)-1 defined? Homework Equations (1-x)-1 = Σ(-x)n= 1-x+x2-x3+... The Attempt at a Solution (1-H)-1 converges if each element of H converges in this series, that is (1-hi)-1...
  22. J

    Proving Isomorphism of Linear Operator with ||A|| < 1

    Hi, I have some trouble with the following problem: Let E be a Banach space. Let A ∈ L(E), the space of linear operators from E. Show that the linear operator φ: L(E) → L(E) with φ(T) = T + AT is an isomorphism if ||A|| < 1. So the idea here is to use the Neumann series but I can't really...
  23. W

    Differential Linear Operator Problem not making sense

    Homework Statement I think there may be something wrong with a problem I'm doing for homework. The problem is: Solve the IVP with the differential operator method: [D^2 + 5D + 6D], y(0) = 2, y'(0) = \beta > 0 a) Determine the coordinates (t_m,y_m) of the maximum point of the solution as a...
  24. rkrishnasanka

    Is pertubation a linear operation?

    My question stems from a discussion I had with my colleague today. In Electomagnetic coupling , like in waveguide structures. We apply pertubation theory to find out the coupling between various modes that get coupled in the device. My colleague said that the coupling interaction was...
  25. L

    MHB Linear operator, its dual, proving surjectivity

    Let T: X \rightarrow Y be a continuous linear operator between Banach spaces. Prove that $T$ is surjective \iff T^* is injective and im T^* is closed. I've proven a "similar" statement, with imT^* replaced with imT. There I used these facts: $\overline{imT}= ^{\perp}(kerT^*)$ and...
  26. D

    Irreducible linear operator is cyclic

    I´m having a hard time proving the next result: Let T:V→V be a linear operator on a finite dimensional vector space V . If T is irreducible then T cyclic. My definitions are: T is an irreducible linear operator iff V and { {\vec 0} } are the only complementary invariant subspaces. T...
  27. W

    Linear operator and linear vector space?

    hi, please tell me what do we mean when we say in quantum mechanics operators are linear and also vector space is also linear ?
  28. G

    Linear operator, linear functional difference?

    What is a difference between linear operator and linear functional? Do I understand it correctly that linear operator is any operator that when applied on a vector from a vector space, gives again a vector from this vector space. And also obeys linearity conditions. And linear functional is a...
  29. DavideGenoa

    Existence of surjective linear operator

    Dear friends, I read that, if ##A## is a bounded linear operator transforming -I think that such a terminology implies that ##A## is surjective because if ##B=A## and ##A## weren't surjective, that would be a counterexample to the theorem; please correct me if I'm wrong- a Banach space ##E##...
  30. A

    Can Any Linear Operator Be Expressed Using Hermitian Components?

    Homework Statement Show that any linear operator \hat{L} can be written as \hat{L} = \hat{A} + i\hat{B}, where \hat{A} and \hat{B} are Hermitian operators. Homework Equations The properties of hermitian operators. The Attempt at a Solution I am not sure where to start with this...
  31. S

    Quotient Space Linear Operator

    Could anyone help me solve this problem? Let A,B be two subspace of V, a \in A, b \in B. Show that the following operation is linear and bijective: (A + B)/B → A/(A \cap B): a + b + B → a + A \cap B I really couldn't understand how the oparation itself works, i.e, what F(v) really is in...
  32. A

    Continuity of the inverse of a linear operator

    If g(a) \neq 0 and both f and g are continuous at a, then we know the quotient function f/g is continuous at a. Now, suppose we have a linear operator A(t) on a Hilbert space such that the function \phi(t) = \| A(t) \|, \phi: \mathbb R \to [0,\infty), is continuous at a. Do we then know that...
  33. A

    Defining the square root of an unbounded linear operator

    I have started coming across square roots (H+kI)^{\frac 12} of slight modifications of Schrodinger operators H on L^2(\mathbb R^d); that is, operators that look like this: H = -\Delta + V(x), where \Delta is the d-dimensional Laplacian and V corresponds to multiplication by some function. But...
  34. T

    Prove that a linear operator is indecomposable

    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...
  35. Sudharaka

    MHB Uniquely Determined Linear Operator

    Hi everyone, :) Here's a problem that I want to confirm my answer. Note that for the second part of the question it states, "prove that \(T\) is bonded by the above claim". I used a different method and couldn't find a method that relates the first part to prove the second. Problem: Suppose...
  36. D

    Finding normalized eigenfunctions of a linear operator in Matrix QM

    Homework Statement Hey everyone! The question is this: Consider a two-state system with normalized energy eigenstates \psi_{1}(x) and \psi_{2}(x), and corresponding energy eigenvalues E_{1} and E_{2} = E_{1}+\Delta E; \Delta E>0 (a) There is another linear operator \hat{S} that acts by...
  37. L

    Interchanging Linear Operator and Infinite Sum

    Suppose that x\in H, where H is a Hilbert space. Then x has an orthogonal decomposition x = \sum_{i=0}^\infty x_i. I have a linear operator P (more specifically a projection operator), and I want to write: P(x) = \sum_{i=0}^\infty P(x_i). How can I justify taking the operator inside the...
  38. D

    Proof of a linear operator acting on an inverse of a group element

    Hey guys! Basically, I was wondering how to prove the following statement. I've seen it in the Hamermesh textbook without proof, so I wanted to know how you go about doing it. Let's say you have a group element g_{1}, which has a corresponding inverse g_{1}^{-1}. Let's also define a linear...
  39. V

    Representation of linear operator using series ?

    representation of linear operator using "series"? I was looking into the progression of quantum states with respect to time. From what I understood the progression of a state ## \left|\psi(t)\right> ## is given by: $$ \left|\psi(t)\right> = U(t)\left|\psi(0)\right> $$ I'm not sure if that's...
  40. A

    Spectrum of a linear operator on a Banach space

    I'm trying to understand the spectrum and resolvent of a linear operator on a Banach space in as much generality as I possibly can. It seems that the furthest the concept can be "pulled back" is to a linear operator T: D(T) \to X, where X is a Banach space and D(T)\subseteq X. But here are a...
  41. DrClaude

    Action of a linear operator on vectors

    Not really a homework problem, just doing some self-studying. Homework Statement Let ##| a \rangle## by any vector in an ##N##-dimensional vector space ##\mathcal{V}##, and ##\mathbf{A}## a linear operator on ##\mathcal{V}##. The vectors $$ | a \rangle, \mathbf{A} | a \rangle...
  42. L

    Proof That Every Linear Operator L:ℝ→ℝ Has Form L(x)=cx

    Homework Statement Show that every linear operator L:ℝ→ℝ has the form L(x) = cx for some c in ℝ. Homework Equations A linear operator in vector space V is a linear transformation whose domain and codomain are both V. The Attempt at a Solution If L is a vector space of the real...
  43. M

    What are the effects of a translation on a vector in R2?

    Homework Statement Let a be a fixed nonzero vector in R2. A mapping of the form L(x) = x + a is called a translation. Show that a translation is not a linear transformation. Illustrate geometrically the effect of a translation. My work is in the photo below, can you check and see if I'm...
  44. M

    Compact linear operator in simple terms?

    Hi, I'm struggling to understand this concept. I think the term probably comes from functional analysis and I don't know any of the terms in that field so I'm having trouble understanding the meaning of what a compact linear operator is. I posted this in linear algebra because I'm reading...
  45. A

    When is the kernel of a linear operator closed?

    If you consider a bounded linear operator between two Hausdorff topological vector spaces, isn't the kernel *always* closed? I mean, if you assume singleton sets are closed, then the set \{0\} in the image is closed, so that means T^{-1}(\{0\}) is closed, right (since T is assumed continuous)? I...
  46. E

    Linear operator or nonlinear operator?

    Homework Statement Verify whether or not the operator L(u) = u_x + u_y + 1 is linear. Homework Equations An operator L is linear if for any functions u, v and any constants c, the property L(c_1 u + c_2 v) = c_1 L(u) + c_2 L(v) holds true. The Attempt at a Solution I feel...
  47. Z

    Spectrum of a linear operator on a Banach space

    Homework Statement I have a number of problems, to be completed in the next day or so (!) that I am pretty stuck with where to begin. They involve calculating the spectra of various different linear operators. Homework Equations The first was: Let X be the space of complex-valued...
  48. N

    Show That g(y)=proj_x y is a Linear Operator.

    Homework Statement Let x be a fixed nonzero vector in R^3. Show that the mapping g:R^3→R^3 given by g(y)=projxy is a linear operator. Homework Equations projxy = \left(\frac{x\cdot y}{\|x\|}\right)x My book defines linear operator as: Let V be a vector space. A linear operator on V is...
  49. T

    Linear algebra problem (standard matrix for a linear operator)

    Homework Statement Determine the standard matrix for the linear operator defined by the formula below: T(x, y, z) = (x-y, y+2z, 2x+y+z) Homework Equations The Attempt at a Solution No idea