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
V
→
W
{\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
V
=
W
{\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
V
{\displaystyle V}
and
W
{\displaystyle W}
are real vector spaces (not necessarily with
V
=
W
{\displaystyle V=W}
), or it can be used to emphasize that
V
{\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.
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...
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...
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...
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...
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...
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...
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...
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...
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}...
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 ...
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...
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...
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 .
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.
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...
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...
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...
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...
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'...
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...
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...
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...
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...
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...
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...
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?
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...
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)...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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