# What is Linear transformation: Definition and 446 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. ### Linear Transformation from R3 to R3

"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case? Thank you.
2. ### Codomain and Range of Linear Transformation

Standard matrix for T is: $$P=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & -1 \end{bmatrix}$$ (i) Since matrix P is already in reduced row echelon form and each row has a pivot point, ##T## is onto mapping of ##\mathbb R^3 \rightarrow \mathbb R^2## (ii) Since there is free variable in matrix P, T is...
3. ### Prove that T is a linear transformation

We got two vectors ##\mathbf{v_1}## and ##\mathbf{v_2}##, their sum is, geometrically, : Now, let us rotate the triangle by angle ##\phi## (is this type of things allowed in mathematics?) OC got rotated by angle ##\phi##, therefore ##OC' = T ( \mathbf{v_1} + \mathbf{v_2})##, and similarly...
4. ### The correct way to write the range of a linear transformation

We have a transformation ##T : V_2 \to V_2## such that: $$T (x,y)= (x,x)$$ Prove that the transformation is linear and find its range. We can prove that the transformation is Linear quite easily. But the range ##T(V_2)## is the the line ##y=x## in a two dimensional (geometrically) space...
5. ### I Dimension of a Linear Transformation Matrix

hi guys I was trying to find the matrix of the following linear transformation with respect to the standard basis, which is defined as ##\phi\;M_{2}(R) \;to\;M_{2}(R)\;; \phi(A)=\mu_{2*2}*A_{2*2}## , where ##\mu = (1 -1;-2 2)## and i found the matrix that corresponds to this linear...
6. ### MHB 072 is Q(theta) a linear transformation from R^2 to itself.

if $Q(\theta)$ is $\left[\begin{array}{rr} \cos{\theta}&- \sin{\theta}\\ \sin{\theta}&\cos{\theta} \end{array}\right]$ how is $Q(\theta)$ is a linear transformation from R^2 to itself. ok I really didn't know a proper answer to this question but presume we would need to look at the unit...
7. ### A Linear transformation problem

I tried hard to understand what this author proposed, but I feel like I failed miserably. My attempt of solution is here: Item (a) is verified in the case where ##n = 2##, since ##F## being a linear transformation, by the Corollary of the Nucleus and Image Theorem, ##F## takes a basis of...
8. ### Linear transformation: Find the necessary quantity of T

> Let ##C## be the disk with radius 1 with center at the origin in ##R^2##. > Consider the following linear transformation: ##T: (x,y) \to (\frac{5x+3y}{4},\frac{3x+5y}{4})## > > What is the lowest number such that ##T^{n}(C)## contains at lest ##2019## points ##(a,b)##, with a and b integers...
9. ### I Trying to get a better understanding of the quotient V/U in linear algebra

Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...
10. ### Question about linear transformations

Summary:: linear transformations Hello everyone, firstly sorry about my English, I'm from Brazil. Secondly I want to ask you some help in solving a question about linear transformations. Here is the question: Consider the linear transformation described by the matrix \mathsf{A} \in \Re...
11. ### I Proving Linear Transformation of V with sin(x),cos(x) & ex

Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A). Define T:V→V,f↦df/dx. How do I prove that T is a linear transformation? (I can do this with numbers but the trig is throwing me).
12. ### MHB Norms for a Linear Transformation .... Browder, Lemma 8.4 ....

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding Lemma 8.4 ... Lemma 8.4 reads as follows...

14. ### I Find matrix of linear transformation and show it's diagonalizable

The strategy here would probably be to find the matrix of ##F##. How would one go about doing that? Since ##V## is finite dimensional, it must have a basis...
15. ### I Proof of ##F## is an orthogonal projection if and only if symmetric

The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is ##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##. In the attached image, second equation, how is the...
16. ### I Understanding linear transformation

How can the function ##F(\mathbf{u})(t)=\mathbf{u}^{(n)}(t)+a_1\mathbf{u}^{(n-1)}(t)+...+a_n\mathbf{u}(t)##, where ##\mathbf{u}\in U=C^n(\mathbf{R})## (i.e. the space of all ##n## times continuously differentiable functions on ##\mathbf{R}##) be a linear transformation (from ##U##) to...
17. ### I Linear transformation T: R3 -> R2

Homework Statement Find the linear transformation [/B] T: R3 --> R2 such that: 𝑇(1,0,−1) = (2,3) 𝑇(2,1,3) = (−1,0) Find: 𝑇(8,3,7) Does any help please?
18. ### MHB 17.1 Determine if T is a linear transformation

nmh{2000} 17.1 Let $T: \Bbb{R}^2 \to \Bbb{R}^2$ be defined by $$T \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 2x+y\\x-4y \end{bmatrix}$$ Determine if $T$ is a linear transformation. So if...
19. ### MHB Help Solving Linear Transformation Problem in R^2

Hey i got a problem here but still without correction so if you guys can help me , thanks in advance I'm stuck there We have L : P -> R^2 L is a linear transformation with : B = \left\{1-x^{2},2x,1+2x+3x^{2} \right\} \; and \; B' = \begin{Bmatrix} \begin{bmatrix} 1\\-1 \end{bmatrix}...
20. ### MHB 13 is a linear transformation and .......Determine T

Suppose that $T: \Bbb{R}^3 \rightarrow \Bbb{R}^3$ is a linear transformation and T \begin{bmatrix} 1 \\1 \\0 \\ \end{bmatrix} = \begin{bmatrix} 1 \\2 \\1 \\ \end{bmatrix}, \quad T \begin{bmatrix} 1 \\0 \\1 \\ \end{bmatrix} = \begin{bmatrix} 1 \\0 \\2 \\ \end{bmatrix}, \quad T...
21. ### Projections and direct sum

Homework Statement Let ##V = \mathbb{R}^4##. Consider the following subspaces: ##V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]## And let ##V = M_n(\mathbb{k})##. Consider the following subspaces: ##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}## ##V_2 =...
22. ### [Linear Algebra] Linear transformation proof

Homework Statement Let ##V## and ##W## be vector spaces, ##T : V \rightarrow W## a linear transformation and ##B \subset Im(T)## a subspace. (a) Prove that ##A = T^{-1}(B)## is the only subspace of ##V## such that ##Ker(T) \subseteq A## and ##T(A) = B## (b) Let ##C \subseteq V## be a...
23. ### [Linear Algebra] Help with Linear Transformation exercises

Homework Statement 1. (a) Prove that the following is a linear transformation: ##\text{T} : \mathbb k[X]_n \rightarrow \mathbb k[X]_{n+1}## ##\text{T}(a_0 + a_1X + \ldots + a_nX^n) = a_0X + \frac{a_1}{2}X^2 + \ldots + \frac{a_n}{n+1}## ##\text{Find}## ##\text{Ker}(T)## and ##\text{Im}(T)##...
24. ### [LinAlg] Show that T:C[a,b] -> R is a linear transformation

Homework Statement Show that T:C[a,b] -> defined by T(f) = ∫(from a to b) f(x)dx is a linear transformation. Homework Equations Definition of a linear transformation: A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique...
25. ### Find the standard matrix of the linear transformation

Homework Statement Homework Equations None. The Attempt at a Solution I know that the standard matrix of a counterclockwise rotation by 45 degrees is: [cos 45 -sin 45] [sin 45 cos 45] =[sqrt(2)/2 -sqrt(2)/2] [sqrt(2)/2 sqrt(2)/2] But the problem says "followed by a projection onto the line...
26. ### I Intuitive Linear Transformation

Is it possible to understand intuitively (without using a formal proof ) why a reflection is a linear function ?
27. ### I Proving the Linear Transformation definition

HI .I'm trying to prove that, for a linear transformation, it is worth that: f(a\bar{x}+b\bar{y})=af(\bar{x})+bf(\bar{y}) for every real numbers a and b. Until now, I have proved by myself that f(\bar{x}+\bar{y})=f(\bar{x})+f(\bar{y}). and , using this result i proved that: f(a\bar{v}) =...
28. ### MHB Norm of a Linear Transformation: Proving Homogeneity From Definition - Peter

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows...
29. ### MHB Help with Proof of Junghenn Proposition 9.2.3 - A Course in Real Analysis

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In the...
30. ### I Norm of a Linear Transformation .... Another question ....

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on ##\mathbb{R}^n##" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In...
31. ### MHB Norm of a Linear Transformation .... Junnheng Proposition 9.2.3 .... ....

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows...
32. ### I Norm of a Linear Transformation .... Junghenn Propn 9.2.3 ....

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on ##\mathbb{R}^n##" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In...
33. ### Finding the Standard Matrix of a Linear Transformation

Homework Statement Let ##T:ℝ^3→ℝ^2## be the linear transformation defined by ##\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\mapsto \begin{bmatrix} x_1 + x_2 + x_3\\ 0 \end{bmatrix}##. i. Find the standard matrix for ##T##. Homework Equations The Attempt at a Solution For this problem I...
34. ### Finding a basis for the linear transformation S(A)=A^T?

Homework Statement "Find ##S_\alpha## where ##S: M_{2×2}(ℝ)→M_{2×2}(ℝ)## is defined by ##S(A)=A^T##. Homework Equations ##A^T=\begin{pmatrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{pmatrix}## ##\alpha= \{ {\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0...
35. ### I Linear transformation of a given coordinate

I have a question about weights of a basis set with respect to the other basis set of one specific vector space. It seems the weights do not covert linearly when basis sets convert linearly. I've got this question from the video on youtube "linear transformation" Let's consider a vector...
36. ### Onto linear transformation

Homework Statement Say I have a matrix: [3 -2 1] [1 -4 1] [1 1 0] Is this matrix onto? One to one? Homework Equations The Attempt at a Solution I know it's not one to one. In ker(T) there are non trivial solutions to the system. But since I've confirmed there is something in the ker(T)...
37. ### Eigenvalues of transpose linear transformation

Homework Statement If ##A## is an ##n \times n## matrix, show that the eigenvalues of ##T(A) = A^{t}## are ##\lambda = \pm 1## Homework Equations The Attempt at a Solution First I assume that a matrix ##M## is an eigenvector of ##T##. So ##T(M) = \lambda M## for some ##\lambda \in...
38. ### Eigenvectors and orthogonal basis

Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...
39. ### I Verifying a linear transformation

I am told that the trace function tr(A) is a linear transformation. But this function maps from the space of matrices to the real numbers. How can this be a linear transformation if the set of real numbers isn't a vector space? Or is it? Can a field also be considered a vector space?
40. ### MHB Find the General Expression for a Linear Transformation

I don't quite get this question, how is it done ?
41. ### Find out if it's linear transformation

Homework Statement Does a linear transformation ##g : \mathbb{R}^2 \rightarrow \mathbb{R}^2## so that ##g((2, -3)) = (5, -4)## and ##g((-\frac{1}{2}, \frac{3}{4})) = (0, 2)## exist? Homework Equations The Attempt at a Solution For a linear transformation to exist we need to know if those two...
42. ### Linear Transformation R4 to R4: KerT + ImT = R4

Homework Statement Let T be a Linear Transformation defined on R4 ---> R4 Is that true that the following is always true ? KerT + ImT = R4 Homework Equations The Attempt at a Solution Since every vector in R4 must be either in KerT or the ImT, so the addition of those subspace contains R...
43. ### Show that the T is a linear transformation

Homework Statement T:R2[x] --> R4[x] T(f(x)) = (x^3-x)f(x^2) Homework Equations The Attempt at a Solution Let f(x) and g(x) be two functions in R2[x]. T(f(x) + g(x)) = T(f+g(x)) = (x^3-x)(f+g)(x^2) = (x^3-x)f(x^2) + (x^3-x)g(x^2) = T(f(x)) + T(g(x)). let a be scalar in R: aT(f(x)) =...
44. ### Finding a matrix for a linear transformation

'Homework Statement Find the matrix A' for T: R2-->R2, where T(x1, x2) = (2x1 - 2x2, -x1 + 3x2), relative to the basis B' {(1, 0), (1, 1)}. Homework Equations B' = {(1, 0), (1, 0)} so B'-1 = {(1, -1), (0, 1)}. The Attempt at a Solution I'm confused at what exactly a transform matrix...
45. ### Linear transformation representation with a matrix

Homework Statement For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}. Homework Equations T(v) is given, (x1+x2, 2x1-x2) The Attempt at a Solution Okay, I see...
46. ### Linear Transformation and Inner Product Problem

Homework Statement Consider the vector space R2 with the standard inner product given by ⟨(a, b), (c, d)⟩ = ac + bd. (This is just the dot product.) PLEASE SEE THE ATTACHED PHOTO FOR DETAIlS Homework Equations T(v)=AT*v The Attempt at a Solution I was able to prove part a. I let v=(v1,v2)...
47. ### I Matrix Representation of Linear Transformation

This is where I am stuck. I studied ordered basis and coordinates vector previous to this. of course I studied vector space, basis, linear... etc too, However I can't understand just this part. (maybe this whole part) Especially this one which says [[T(b1)]]c...[[T(bn)]]c be a columns of...
48. ### MHB How to define this linear transformation

> Admit that $V$ is a linear space about $\mathbb{R}$ and that $U$ and $W$ are subspaces of $V$. Suppose that $S: U \rightarrow Y$ and $T: W \rightarrow Y$ are two linear transformations that satisfy the property: > $(\forall x \in U \cap W)$ $S(x)=T(x)$ > Define a linear transformation $F$...
49. ### I How this defines a linear transformation

Admit that V is a linear space about \mathbb{R} and that U and W are subspaces of V. Suppose that S: U \rightarrow Y and T: W \rightarrow Y are two linear transformations that satisfy the property: (\forall x \in U \cap W) S(x)=T(x) Define a linear transformation F: U+W \rightarrow Y that...
50. ### When is this linear transformation an isomorphism?

Homework Statement Let L: ℝ2→ℝ2 such that L(x1, x2)T=(1, 2 ; 3, α)(x1, x2)T=Ax Determine at what values of α is L an isomorphism. Obviously L is given in matrix form. The Attempt at a Solution First of all a quick check, dim (ℝ2)=dim(ℝ2)=2 Ok. An isomorphism means linear transformation which...