What is linear operators: Definition and 61 Discussions

In mathematics, and more specifically in linear algebra, 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. nomadreid

    I Want to understand how to express the derivative as a matrix

    In https://www.math.drexel.edu/~tolya/derivative, the author selects a domain P_2 = the set of all coefficients (a,b,c) (I'm writing horizontally instead off vertically) of second degree polynomials ax^2+bx+c, then defines the operator as matrix to correspond to the d/dx linear transformation...
  2. cookiemnstr510510

    Linear operators, quantum mechanics

    Hello, I am struggling with what each piece of these equations are. I generally know the two rules that need to hold for an operator to be linear, but I am struggling with what each piece of each equation is/means. Lets look at one of the three operators in question. A(f(x))=(∂f/∂x)+3f(x) I...
  3. S

    MHB On the spectral radius of bounded linear operators

    Hi EVERYBODY: General knowledge: The homogeneous linear Fredholm integral equation $\mu\ \varPsi(x)=\int_{a}^{b} \,k(x,s) \varPsi(s) ds$ (1) has a nontrivial solution if and only if $\mu$ is an eigenvalue of the integral operator $K$. By multiplying (1) by $k(x,s)$ and...
  4. Whiteboard_Warrior

    I Parameterization of linear operators on the holomorphisms

    Let D represent the differentiation of a single-parameter holomorphism, with respect to its parameter x. It's clear that for any sequence of holomorphisms g on x, sigma[k=0,inf](g[k](x)*D^k) is a linear operator on the space of holomorphisms. Is this a complete parameterization of the linear...
  5. 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...
  6. pellman

    I Aren't all linear operators one-to-one and onto?

    Let W be a vector space and let A be a linear operator W --> W. Isn't it the case that for any such A, the kernel of A is the zero vector and the range is all of W? And that it is one-to-one from linearity? I ask because an author I am reading goes through a lot of steps to show that a certain...
  7. V

    Linear Algebra - Linear Operators

    Homework Statement True or false? If T: ℙ8(ℝ) → ℙ8(ℝ) is defined by T(p) = p', so exists a basis of ℙ8(ℝ) such that the matrix of T in relation to this basis is inversible. Homework EquationsThe Attempt at a Solution So i think that my equations is of the form: A.x = x' hence A is...
  8. A

    Is Matrix Addition Commutative?

    Suppose we have linear operators A' and B'. We define their sum C'=A'+B' such that C'|v>=(A'+B')|v>=A'|v>+B'|v>. Now we can represent A',B',C' by matrices A,B,C respectively. I have a question about proving that if C'=A'+B', C=A+B holds. The proof is Using the above with Einstein summation...
  9. D

    Linear operators and vector spaces

    Hi all, I've been doing some independent study on vector spaces and have moved on to looking at linear operators, in particular those of the form T:V \rightarrow V. I know that the set of linear transformations \mathcal{L}\left( V,V\right) =\lbrace T:V \rightarrow V \vert \text{T is linear}...
  10. D

    Linear operators and change of basis

    Following on from a previous post of mine about linear operators, I'm trying to firm up my understanding of changing between bases for a given vector space. For a given vector space V over some scalar field \mathbb{F}, and two basis sets \mathcal{B} = \lbrace\mathbf{e}_{i}\rbrace_{i=1,\ldots ...
  11. D

    Linear operators & mappings between vector spaces

    Hi, I'm having a bit of difficulty with the following definition of a linear mapping between two vector spaces: Suppose we have two n-dimensional vector spaces V and W and a set of linearly independent vectors \mathcal{S} = \lbrace \mathbf{v}_{i}\rbrace_{i=1, \ldots , n} which forms a basis...
  12. H

    Showing determinant of product is product of dets for linear operators

    Homework Statement Assume A and B are normal linear operators [A,A^{t}]=0 (where A^t is the adjoint) show that det AB = detAdetB Homework Equations The Attempt at a Solution Well I know that since the operators commute with their adjoint the eigenbases form orthonormal sets...
  13. T

    MHB Prove the following; (vector spaces and linear operators)

    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 .
  14. F

    MHB Convergence of bounded linear operators

    Let (T_{n}) be a sequence in {B(l_2} given by T_{n}(x)=(2^{-1}x_{1},...,2^{-n}x_{n},0,0,...). Show that T_{n}->T given by T(x)==(2^{-1}x_{1},2^{-2}x_{2},0,0,...). I get a sequence of geometric series as my answer for the norm, but not sure whether that's correct.
  15. fluidistic

    How Does the Adjoint of a Linear Operator Work in Hilbert Spaces?

    Homework Statement I must show several properties about linear operators using the definition of the adjoint operator. A and B are linear operator and ##\alpha## is a complex number. The first relation I must show is ##(\alpha A + B)^*=\overline \alpha A^*+B^*##. Homework Equations The...
  16. E

    A question on the product of two real linear operators

    I am reading The Principles of Quantum Mechanics 4th Ed by Paul Dirac, specifically where he introduces his own Bra-Ket notation. You can view this book as a google book. http://books.google.com.au/books?id=XehUpGiM6FIC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false...
  17. M

    Is A=5 and A=x Linear Operators?

    Linear operator A is defined as A(C_1f(x)+C_2g(x))=C_1Af(x)+C_2Ag(x) Question. Is A=5 a linear operator? I know that this is just number but it satisfy relation 5(C_1f(x)+C_2g(x))=C_15f(x)+C_25g(x) but it is also scalar. Is function ##A=x## linear operator? It also satisfy...
  18. Roodles01

    What is the role of linear operators in quantum mechanics?

    Homework Statement Just starting third level Uni. stuff & am faced with linear operators from Quantum Mechanics & need a little help. OK, an operator, Ô, is said to be linear if it satisfies the equation Ô(α f1 + β f2) = α(Ô f1) + β(Ô f2) Fine but I have an equation I can't wrap my...
  19. D

    Quantum Physics - hermitian and linear operators

    Description 1. Prove that operators i(d/dx) and d^2/dx^2 are Hermitian. 2. Operators A and B are defined by: A\psi(x)=\psi(x)+x B\psi(x)=d\psi/dx+2\psi/dx(x) Check if they are linear. The attempt at a solution I noted the proof of the momentum operator '-ih/dx'...
  20. C

    Linear Operators and Dependence

    Suppose that T: V →V is a linear operator and {v1, . . . , vn} is linearly dependent. Show that {T (v1), . . . , T (vn)} is linearly dependent. I'm pretty lost as to how to even go about doing this problem, but I'll take a crack at it. I'm not sure what the operator "T:V→V" means. It...
  21. S

    Linear Operators: Are They Inverse & Adjoint?

    Homework Statement Consider the following operators acting in the linear space of functions Ψ(x) defined on the interval (∞,∞) (a) Shift Ta: TaΨ(x)=Ψ(x+a), a is a constant (b) Reflection (inversion) I: IΨ(x)=Ψ(x) (c) Scaling Mc: McΨ(x)= √c Ψ(cx), c is a constant (d) Complex conjugation K...
  22. M

    What is the suitable representation of a linear operator of matrices?

    Hi there, As you know, we can represent a Linear vector operator as a matrix product, i.e., if T(u) = v, there is a matrix A that u = A.v. What about a linear operator of matrices. I have a T(X) = b where X belongs to R^n_1Xn_2 and b belongs to R^p. What is a suitable representation of...
  23. A

    What is the spectrum of A in terms of S and its eigenvalues?

    Homework Statement Let A be a linear transformation on the space of square summable sequences \ell2 such that (A\ell)n = \elln+1 + \elln-1 - 2\elln. Find the spectrum of A. 2. The attempt at a solution I see that A is self-adjoint, so its spectrum must be a subset of the real line. We also...
  24. T

    Linear Algebra (Matrix representation of linear operators)

    Homework Statement Determine [T]β for linear operator T and basis β T:((x1; x2]) = [2x1 + x2; x1 - x2] β = {[2; 1], [1; 0]} Homework Equations Now that would be MY question :rolleyes: The Attempt at a Solution Well the answer is [1, 1; 3, 0], but i have no idea what I'm even...
  25. A

    Densely defined linear operators on Hilbert space and their ranges

    Suppose T is an injective linear operator densely defined on a Hilbert space \mathcal H. Does it follow that \mathcal R(T) is dense in \mathcal H? It seems right, but I can't make the proof work... There is a theorem that speaks to this issue in Kreyszig, and also in the notes provided by my...
  26. A

    Question about linear operators

    Apparently - that is, if I'm to believe Kolmogorov - we have the following for a bounded linear operator A between two normed spaces: \sup_{\| x \| \leq 1} \|Ax\| = \sup_{\|x\| = 1} \|Ax\| But why?
  27. D

    Diagonalizing Linear Operators: Understanding the Differences

    Homework Statement Let V be a n-dimensional real vector space and L: V --> V be a linear operator. Then, A.) L can always be diagonalized B.) L can be diagonalized only if L has n distinct eigenvalues C.) L can be diagonalized if all the n eigenvalues of L are real D.) Knowing the...
  28. S

    Proving Non-Linearity of y2: Linear Operators Homework

    Homework Statement Show that y2 is non-linear. Homework Equations ^O (ay1 +by2) = a ^O(y1) + b ^O(y2). The Attempt at a Solution No idea!
  29. I

    How Do You Determine Linear Transformations in R^2?

    Homework Statement If L((1,2)^T) = (-2,3)^T and L((1, -1)T) = (5,2)T determine L((7,5)T) Homework Equations If L is a linear transformation mapping a vector V into W, it follows: L(v1 + v2) = L(v1) +L(v2) (alpha = beta = 1) and L (alpha v) = alpha L(v) (v = v1, Beta = 0)...
  30. G

    Is B(X,Y) a Vector Space of Bounded Linear Operators over the Same Scalar Field?

    Homework Statement Show L(X,Y) is a vector space. Then if X,Y are n.l.s. over the same scalar field define B(X,Y) = set of all bounded linear operators for X and Y Show B(X,Y) is a vector space(actually a subspace of L(X,Y) Homework Equations The Attempt at a Solution im not sure if i have...
  31. A

    Linear Operators: Relationship Between Action on Kets & Bras

    This is basically more of a math question than a physics-question, but I'm sure you can answer it. My question is about linear operators. If I write an operator H as (<al and lb> being vectors): <alHlb> What is then the relationship between H action the ket and H action on the bra. Is this for...
  32. M

    Commutative linear operators and their properties

    Can someone help me with this? When two linear operators commute, I know how to show that they must have at least one common eigenvector. Beyond this fact, what else can be said about commutative operators and their eigenvectors? Further, can they be diagonalized simultaneously (or actually, can...
  33. R

    Pervasiveness of linear operators

    Obviously linear operators are ideal to work with. But is there a deeper reason explaining why they're ubiquitous in quantum mechanics? Or is it just because we've constructed operators to be linear to make life easier?
  34. A

    Linear Operators: Identifying and Solving

    Homework Statement I have some operators, and need to figure out which ones are Linear (or not). For example: 1. \hat{A} \psi(x) \equiv \psi(x+1) Homework Equations I have defined the Linear Operator: \hat{A}[p\psi_{1}+q\psi_{2}]=p\hat{A}\psi_{1}+q\hat{A}\psi_{2} The Attempt at a...
  35. J

    Linear Algebra with linear operators and rotations

    Definie linear operators S and T on the x-y plane as follows: S rotates each vector 90 degress counter clockwise, and T reflects each vector though the y axis. If ST = S o T and TS = T o S denote the composition of the linear operators, and I is the indentity map which of the following is true...
  36. D

    Proving Linearity of Matrix Operators: Is L(A)=2A a Linear Operator?

    L(A)=2A My book doesn't have any examples of how to do this with matrices so I don't know how to approach this.
  37. B

    Linear operators, eigenvalues, diagonal matrices

    So I have a couple of questions in regards to linear operators and their eigenvalues and how it relates to their matrices with respect to some basis. For example, I want to show that given a linear operator T such that T(x_1,x_2,x_3) = (3x_3, 2x_2, x_1) then T can be represented by a diagonal...
  38. E

    Linear Operators in Hilbert Space - A Dense Question

    Let H be a Hilbert space and let S be the set of linear operators on H. Is there a proper subset of S that is dense in S?
  39. Z

    Can Linear Operators A and B Affect the Rank of AB in V?

    Studying old exam papers from my college I came across the following: Given linear operators A,\,B: V\rightarrow V, show that: \textrm{rk}AB\le \textrm{rk}A My solution: Since all v \in \textrm{Ker}B are also in \textrm{Ker}AB (viz ABv=A(Bv)=A(0)=0) and potentially there are w \in...
  40. N

    Linear Operators and Eigen Values

    I'm looking for a good website for understanding Quantum Mechanics (i.e. Time Independent Schrodinger Eq'n, Harmonic Oscillators, Rigid Rotors, etc) The operator is linear if the following is satisfied: A[c*f(x)+d*g(x)]=c*A[f(x)]+d*A[fg(x)], where A = an operator of any kind I'm having...
  41. F

    What is the equation for representing a linear operator in terms of a matrix?

    I'm working through a proof that every linear operator, A, can be represented by a matrix, A_{ij}. So far I've got which is fine. Then it says that A(\textbf{e}_{i}) is a vector, given by: A(e_{i}) = \sum_{j}A_{j}(p_{i})e_{j} = \sum_{j}A_{ji}e_{j}. The fact that its a vector is fine...
  42. Z

    Dimension of a subspace of polynomials with certain coefficients

    Right so I've had an argument with a lecturer regarding the following: Suppose you consider P_4 (polynomials of degree at most 4): A(t)=a_0+a_1t+a_2t^2+a_3t^3+a_4t^4 Now if we consider the subspace of these polynomials such that a_0=0,\ a_1=0,\ a_2=0}, I propose that the dimension of of this...
  43. S

    About the invariance of similar linear operators and their minimal polynomial

    About the invariance of similar linear operators and their minimal polynomial Notations: F denotes a field V denotes a vector space over F L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is...
  44. S

    About the linear dependence of linear operators

    Notations: F denotes a field V denotes a vector space over F L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is composition of functions. τ denotes a linear operator contained in L(V) ι...
  45. 1

    The subtle difference between matrices and linear operators

    For example, if I were to prove that all symmetric matrices are diagonalizable, may I say "view symmetric matrix A as the matrix of a linear operator T wrt an orthonormal basis. So, T is self-adjoint, which is diagonalizable by the Spectral thm. Hence, A is also so." Is it a little awkward to...
  46. L

    Eigenvalues of linear operators

    Let V be the vector space of all real integrable functions on [0,1] with inner product <f,g>=\int_0^1 f(t)g(t)dt Three linear operators defined on this space are A=d/dt and B=t and C=1 so that Af=df/dt and Bf=tf and Cf=f I need to find the eigenvalues of these operators: For A...
  47. D

    Explore Banach Spaces and Bounded Linear Operators

    Homework Statement http://img252.imageshack.us/img252/4844/56494936eo0.png 2. relevant equations BL = bounded linear space (or all operators which are bounded). The Attempt at a Solution I got for the first part: ||A||_{BL} =||tf(t)||_{\infty} \leq ||f||_{\infty} so ||A||_{BL} \leq 1...
  48. J

    Invariant subspaces under linear operators

    Homework Statement Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V.Homework Equations U is invariant under a linear operator T if u in U implies T(u) is in U.The Attempt at a Solution Assume {0} does not equal U does not...
  49. W

    Linear operators and a change of basis

    So...I've got an operator. Omega = (i*h-bar)/sqrt(2)[ |2><1| + |3><2| - |1><2| - |2><3| ] Part a asks if this is Hermitian, and my answer, unless I'm missing something, is no. Because the second part in square brackets is |1><2| + |2><3| - |2><1| - |2><3| which is not the same as Omega...
  50. M

    Linear algebra+ linear operators

    Homework Statement In R^{3} ||x||= a_{1}*|x_{1}|+ a_{2}*|x_{2}|+ a_{3}*|x_{3}|. where a_{i}>0 What is ||A||(indused norm = sup||Ax|| as ||x||=1). (Suppose we know the coeffisients of the matrix/operator A)?? Homework Equations The Attempt at a Solution