# 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

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.

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1. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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?

43. ### 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...
44. ### 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...
45. ### 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...
46. ### 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...
47. ### 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...
48. ### 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...
49. ### 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...
50. ### 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