What is Linear map: Definition and 58 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

    Operator T, ##T^2=I##, -1 not an eigenvalue of T, prove ##T=I##.

    Now, for ##v\in V##, ##(T+I)v=0\implies Tv=-v##. That is, the null space of ##T+I## is formed by eigenvectors of ##T## of eigenvalue ##-1##. By assumption, there are no such eigenvectors (since ##-1## is not an eigenvalue of ##T##). Hence, if ##(T-I)v \neq 0## then ##(T+I)(T-I)v\neq 0##...
  2. S

    I A = norm-preserving linear map (+other conditions) => A = lin isometry

    I'm studying "Semi-Riemannian Geometry: The Mathematical Langauge of General Relativity" by Stephen Newman. Theorem 4.4.4 in that book: The proof of part 2 is given like this: Seems a bit incomplete. I'd like to know if my approach is correct: $$\langle A(v+tw),A(v+tw)\rangle=\langle...
  3. K

    Linear algebra, find a basis for the quotient space

    Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...
  4. M

    MHB Kernel of Linear Map: Show $\ker \phi$ Equation

    Hey! :o Let $1\leq n,m\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $(b_1, \ldots , b_n)$ a basis of $V$. Let $W:=\mathbb{R}^m$ and let $\phi:V\rightarrow W$ be a linear map. Show that $$\ker \phi =\left \{\sum_{i=1}^n\lambda_ib_i\mid \begin{pmatrix}\lambda_1\\ \vdots \\ \lambda_n\end{pmatrix}\in...
  5. Fochina

    Proof: Relationship between a linear map and the associated matrix

    Hi! I don't understand how to demonstrate the following exercise. Let ##F: R^{n} \rightarrow R^{n}## be a linear map which is invertible. Show that if ##A## is the matrix associated with ##F##, then ##A^{-1}## is the matrix associated with the inverse of ##F##.
  6. M

    MHB Give a linear map that satisfies given properties

    Hey! :o Let $v_1:\begin{pmatrix}1 \\ 1\\ 1\end{pmatrix}, \ \ v_2:\begin{pmatrix}1 \\ 0\\ 1\end{pmatrix}\in \mathbb{R}^3$. Let $w=\begin{pmatrix}1 \\ 0 \\2\end{pmatrix}\in \mathbb{R}^3$. If possible, give a linear map $\phi:\mathbb{R}^3\rightarrow \mathbb{R}^2$ such that $\phi...
  7. J

    MHB Show the Linear Map is Diagonalizable

    Problem: Suppose $V$ is a complex vector space of dimension $n$, where $n > 0$, and suppose that $T$ is a linear map from $V$ to $V$. Suppose that if $\lambda$ is any eigenvalue of $T$, then $ker(T−\lambda I)^2 = ker(T−\lambda I)$. Prove that $T$ is diagonalizable. Here's what I think I need...
  8. M

    MHB Kernel of Linear Map $\theta$ in $\mathbb{F}_{q^n}$

    Hey! :o Let $q$ be a power of a prime and $n\in \mathbb{N}$. We symbolize with $Tr$ the map of the trace from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$, i.e. $Tr:\mathbb{F}_{q^n}\rightarrow \mathbb{F}_q$, $\displaystyle{Tr(a)=\sum_{j=0}^{n-1}a^{q^j}}$. I want to calculate the dimension of the image...
  9. J

    MHB Linear Map, T^p(x)=0, Show Linear Independence

    Problem: Suppose V is a complex vector space of dimension n, and T is a linear map from V to V. Suppose $x \in V$, and p is a positive integer such that $T^p(x)=0$ but $T^{p-1}(x)\ne0$. Show that $x, Tx, T^2x, ... , T^{p-1}x$ are linearly independent.During class my professor said it was "a...
  10. S

    I Linear mapping of a binary vector based on its decimal value

    Given an ##N## dimensional binary vector ##\mathbf{v}## whose conversion to decimal is equal to ##j##, is there a way to linearly map the vector ##\mathbf{v}## to an ##{2^N}## dimensional binary vector ##\mathbf{e}## whose ##(j+1)##-th element is equal to ##1## (assuming the index starts...
  11. jamalkoiyess

    I Does this theorem need that Ker{F}=0?

    I have encountered this theorem in Serge Lang's linear algebra: Theorem 3.1. Let F: V --> W be a linear map whose kernel is {O}, then If v1 , ... ,vn are linearly independent elements of V, then F(v1), ... ,F(vn) are linearly independent elements of W. In the proof he starts with C1F(v1) +...
  12. Zero2Infinity

    Check of a problem about nullspace

    Homework Statement Let ##V\subset \mathbb{R}^3## be the subspace generated by ##\{(1,1,0),(0,2,0)\}## and ##W=\{(x,y,z)\in\mathbb{R}^3|x-y=0\}##. Find a matrix ##A## associated to a linear map ##f:\mathbb{R}^3\rightarrow\mathbb{R}^3## through the standard basis such that its nullspace is ##V##...
  13. Zero2Infinity

    Write a matrix given the null space

    Homework Statement Build the matrix A associated with a linear transformation ƒ:ℝ3→ℝ3 that has the line x-4y=z=0 as its kernel. Homework Equations I don't see any relevant equation to be specified here . The Attempt at a Solution First of all, I tried to find a basis for the null space by...
  14. F

    I Difference between 'Field' (algebra) and 'Field' (geometry)

    I am trying to build up a kind of mind map of the following: Module (eg. vector space) Ring (eg Field) Linear algebra (concerning vectors and vector spaces, from what I understood) Multilinear Algebra (analogously concerning tensors and multi-linear maps) Linear maps & Multilinear maps The...
  15. M

    Proof of Invertibility: Linear Map's Surjectivity and Injectivity Condition

    I am trying to understand the following basic proposition about invertibility: a linear map is invertible if and only if it is injective and surjective. Now suppose ##T## is a linear map ##T:V\rightarrow W##. The book I read goes the following way in proving the proposition in the direction when...
  16. D

    Adjoint of an adjoint of a linear map

    My question is as it says in the title really. I've been reading Nakahara's book on geometry and topology in physics and I'm slightly stuck on a part concerning adjoint mappings between vector spaces. It is as follows: Let W=W(n,\mathbb{R}) be a vector space with a basis...
  17. Petrus

    MHB Linear Map Input: Solving P'(1-x) | \pi\rangle

    Hello! I have hard to understand this input for this linear map T:P_3(R)->P_2(R) T(p(x))=P'(1-x) so they get this value when they put in which I have hard understanding I don't understand how they get those, I am totally missing something basic...! The only logical explain is that in p'(x)=3x^2...
  18. T

    Is There a Linear Transformation to Map Data Set X to Y in PCA?

    This question broadly relates to principle component analysis (PCA) Say you have some data vector X, and a linear transformation K that maps X to some new data vector Z: K*X → Z Now say you have another linear transformation P that maps Z to a new data vector Y: P*Z → Y is there...
  19. Barioth

    MHB Find the eigenvalue of a linear map

    Hi everyone, I have this linear map A:R^3 \rightarrow R^3 I have that A(v)=v-2(v\dot ô)ô); v,ô\in R^3 ;||ô||=1 I know that A(A(v))=v this telling me that A is it's own inverse. From there, how can I find the eigenvalue of A? Thanks
  20. M

    Linear algebra-Basis of a linear map

    Homework Statement Let ##L: R^{2} → R^{2}## be a linear map such that ##L ≠ O## but## L^{2} = L \circ L = O.## Show that there exists a basis {##A##, ##B##} of ##R^{2}## such that: ##L(A) = B## and ##L(B) = O.## The Attempt at a Solution Here's the...
  21. Fernando Revilla

    MHB Lisa's question at Yahoo Answers (Matrix of a linear map)

    Here is the question: Here is a link to the question: Linear Algebra Problem *Help Please*? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  22. Fernando Revilla

    MHB Surjective and injective linear map

    I quote an unsolved question from MHF posted by user jackGee on February 3rd, 2013. P.S. Of course, I meant in the title and instead of an.
  23. Fernando Revilla

    MHB Linear Map Problem: Proving Equivalence of a & b

    I quote an unsolved problem from MHF (Linear map problem) posted by user jdm900712
  24. J

    Linear Map Problem: Proving a and b Equivalent

    Homework Statement Let V be a vector space over the field F. and T \in L(V, V) be a linear map. Show that the following are equivalent: a) I am T \cap Ker T = {0} b) If T^{2}(v) = 0 -> T(v) = 0, v\in V Homework Equations The Attempt at a Solution Using p -> (q -> r) <->...
  25. S

    Finding rank and nullity of a linear map.

    Homework Statement let a be the vector [2,3,1] in R3 and let T:R3-->R3 be the map given by T(x) =(ax)a State with reasons, the rank and nullity of THomework Equations The Attempt at a Solution Im having trouble understanding this... I know how to do this with a matrix ie row reduce and no. of...
  26. H

    Showing differentiation is a linear map

    Homework Statement The Attempt at a Solution For part ii) I wrote it out as a matrix, getting \begin{array}{ccccccc} 0 & 0 & 0 & 0 & ... & 0 \\ 0 & 0 & 2 & 0 & ... & 0 \\ 0 & 0 & 0 & 6 & ... & 0 \\ . & . & . & . & . & . \\ 0 & 0 & 0 & 0 & ... & N(N-2) \end{array} So...
  27. N

    What does this notation mean? Linear map A = [A^\mu \nu]_\mu \nu

    It's used in a certain proof that I'm reading. A is a linear map from a vectorspace V onto itself. They say they can rewrite the vector space as \mathcal V = \bigoplus_\mu \mathbb C^{m_\mu} \otimes \mathcal V^\mu and I understand this, but they then claim one can (always, as any linear map)...
  28. jinksys

    Verify that any square matrix is a linear operator when considered as a linear map.

    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...
  29. S

    Eigenvalues of a linear map over a finite field

    Homework Statement Let F be a finite field of characteristic p. As such, it is a finite dimensional vector space over Z_p. (a) Prove that the Frobenius morphism T : F -> F, T(a) = a^p is a linear map over Z_p. (b) Prove that the geometric multiplicity of 1 as an eigenvalue of T is 1. (c) Let F...
  30. A

    If f(x) = 0 for every bounded linear map f, is x = 0?

    Suppose you're looking at a complex vector space X, and you know that, for some x in X, you have f(x) = 0 for every linear map on X. Can you conclude that x = 0? If so, how? This seems easy, but I can't think of it for some reason. (EDIT: Assume it holds for every CONTINUOUS (i.e...
  31. H

    Linear map from n-dim space to p-dim space

    I've been thinking about the following question: if x\in R^n and y=Cx\in R^p where matrix C describes the linear map from n dimensional reals to p dimensional reals. If we only have access to y and want to recover the information about x, which components in x are needed? I kind of figured out...
  32. M

    Is this statement about the rank of a linear map true or false?

    Is this statement true or false if false a counterexample is needed if true then an explanation If T : U \rightarrow V is a linear map, then Rank(T) \leq (dim(U) + dim(V ))/2
  33. R

    Proving Linear Map f is a Tensor of Type (1,1)

    Homework Statement Let V and W be vector spaces and let f:V\rightarrow W be a linear map. Show that f is a tensor of type (1,1) Can someone please show how to do this , I have no idea how to do it. Homework Equations The Attempt at a Solution
  34. M

    Is V Isomorphic to R^2 Under the Given Mapping?

    Homework Statement Let V={a cosx + b sinx | a,b \in R} (a) Show that V is a subspace of the R-vector space of all maps from R to R. (b) Show that V is isomorphic to R^2, under the map f: V\rightarrowR^2 a cosx + b sinx \rightleftharpoons [ a over b ] (this is...
  35. M

    Show that a linear map is linearly independent

    Homework Statement Let f:V\rightarrow V be a linear map and let v\inV be such that f^n(v)\neq0 and f^(n+1)(v)=0. Show that v,f(v),...,f^(n-1)(v) are linearly independent. The Attempt at a Solution I'm really stuck with this one. I know the definition of linear independence and...
  36. A

    Understanding Singular Linear Maps: R^m -> R^n

    Homework Statement the question here said is L, linear transformation/mapping is singular? i'm still googling the definition singular linear map, can anyone give me the definition please T_T p/s; i thought it L maybe the matrix representation, but the question L : R^m -> R^n...
  37. T

    What is the Image of a Plane Under a Linear Transformation?

    Homework Statement Let T: \mathbb{R}^3 \to \mathbb{R}^3 be the linear map represented by the matrix \begin{pmatrix} 4 & -1 & 0 \\ 6& 3 & -2\\ 12& 6 & -4\end{pmatrix} What is the image under T of the plane 2x - 5y + 2z = -5? Homework Equations None The Attempt at a Solution I...
  38. 1

    How to Determine the Matrix of a Linear Map with a Non-Standard Basis?

    Homework Statement T(2,1)---> (5,2) and T(1,2)--->(7,10) is a linear map on R^2. Determine the matrix T with respect to the basis B= {(3,3),(1,-1)} Homework Equations The Attempt at a Solution matrix = 5 7 2 10 ?
  39. P

    Finding basis for kernal of linear map

    Homework Statement Let A = 1 3 2 2 1 1 0 -2 0 1 1 2 Viewing A as a linear map from M_(4x1) to M_(3x1) find a basis for the kernal of A and verify directly that these basis vectors are indeed linearly independant. The Attempt at a...
  40. J

    Diagonalisation of a linear map

    For the theorem: " If v1,...,vr are eigenvectors of a linear map T going from vector space V to V, with respect to distinct eigenvalues λ1,...,λr, then they are linearly independent eigenvectors". Are the λ-eigenspaces all dimension 1. for each λ1,...,λr.? Is the dimension of V, r? ie...
  41. C

    Dot product of vector and symmetric linear map?

    Homework Statement My book states as follows: --- If u and v have the coordinate vectors X and Y respectively in a given orthonormal basis, and the symmetric, linear map \Gamma has the matrix A in the same basis, then \Gamma(u) and \Gamma(v) have the coordinates AX and AY, respectively. This...
  42. K

    Linear Map w/ Matrix: Solve for a + d

    Homework Statement Consider the map L from the space of 2x2-matrices to R given by: L([a b]) = a+ d ([c d]) For clarity, that's L(2x2 matrix) = a + d The Attempt at a Solution Im confused how any function of a matrix could possibly equal addition of two scalars, and thus have no...
  43. H

    Field of modulo p equiv classes, how injective linear map -> surjectivity

    Field of modulo p equiv classes, how injective linear map --> surjectivity Homework Statement Let Fp be the field of modulo p equivalence classes on Z. Recall that |Fp| = p. Let L: Fpn-->Fpn be a linear map. Prove that L is injective if and only if L is surjective. Homework Equations...
  44. H

    Matrix A for Linear Map T: R3→R3

    Homework Statement Determine the matrix A for the linear map T: R3→R3 which is defined by that the vector u first is mapped on v×u, where v=(-9,2,9) and then reflected in the plane x=z (positively oriented ON-system). Also determine the determinant for A. Homework Equations The...
  45. B

    Linear Map = Function of degree P-1

    If p is prime, prove that for every function f: Fp -> Fp there exists a polynomial Q (depending on f) of degree at most p-1 such that f(x) = Q(x) for each x in Fp.
  46. S

    Basis for the image of a surjective linear map.

    Homework Statement Let V and W be vector spaces over F, and let T: V -> W be a surjective (onto) linear map. Suppose that {v1, ..., v_m, u1, ... , u_n} is a basis for V such that ker(T) = span({u1, ... , u_n}). Show that {T(v1), ... , T(v_m)} is a basis for W. Homework Equations Basic...
  47. M

    Discovering the Image of a Linear Map: Methods and Examples

    Hi all! Does anyone know a general method for determining the image of a lin map? I´m aware of the definition if im, but how could I determine it. Maybe it would be useful to show this on some examples :)
  48. M

    Suppose T is a linear map and dim(Im(T))=k

    Please, help me! Suppose T is a linear map and dim(Im(T))=k. Prove that T has at most k+1 distinct eigenvalues. Thank you in advance!
  49. J

    Prove "dim U \geq dim V - dim W" Linear Map Question

    Homework Statement Suppose that V and W are finite dimensional and that U is a subspace of V. Prove that there exists T \in L(V,W) such that null T = U if and only if dim U \geq dim V - dim W. Homework Equations thm: If T \in L(V,W), then range T is a subspace...
  50. P

    Linear Map f:R^2 to R^3 with Given Inputs (1,2) and (2,1)

    Homework Statement Find the linear map f:R^2 \rightarrow R^3, with f(1,2) = (2,1,0) and f(2,1)=(0,1,2) Homework Equations The Attempt at a Solution I actually don't understand this task. PLease help! Thank you...