# What is Linear transformations: Definition and 200 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. ### I Conjugation vs Change of Basis

For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix. For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself. The expressions have similar form except for the order of the inverses. Is there there any connection...
2. ### 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...
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. ### 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.
5. ### I Angle-Preserving Linear Transformations in 2D Space for Relativity

I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the ##ct## axis and the worldline of an observer moving relative to me will be at some angle w.r.t. the ##y## axis. When we switch...
6. ### MHB 5.2a plot linear transformations

ok we are supposed to go to here Find 3 different matrices that reflect the following transformations, report the matrix, the determinant, and the eigenvalues. 1. Rotation by $\dfrac{\pi}{4}$ 2. Shear along $x$ by a factor of $k$ 3. Reflection by the line $\theta$ there are some more but the...
7. ### I Given two linear transformations L and K, show ##K = \lambda L## holds

Let ##V## be a real vectorspace of finite dimension ##n##. Let ##L, K:V \rightarrow \Re## be linear transformations so that ##ker(L) \subset ker(K)##. Then there's a parameter ##\lambda \in \Re## so that ##K=\lambda L## a) Show that ##K=\lambda L## holds when ##K=0##. b) Suppose that ##K...
8. ### MHB Linear Transformations & Matrices: Armstrong, Tapp Chs. 9 & 1 - Explained

At the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry" (see text below) writes the following: " ... ... Each matrix A in this group determines an invertible linear transformation f_A: \mathbb{R} \to \mathbb{R} defined by f_A(x) = x A^t ... ... "I know that one may define...
9. ### 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...
10. ### 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).
11. ### MHB Understanding Browder's Remarks on Linear Transformations

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 8: Differentiable Maps ... ... and am currently focused on Section 8.1 Linear Algebra ... ... I need some help in order to fully understand some remarks by Browder in Section 8.1, page...
12. ### I Find the image of a projection

## \dfrac{1}{2} \left(\begin{array}{rrrr} 0 & 2 & -2 & 0 \\ 3 & -2 & 2 & -3 \\ 3 & -4 & 4 & -3 \\ -2 & 2 & -2 & 2 \end{array}\right)## The matrix satisfies ##A^2=A##, so it is a projection. To find ##U'##, one can find the ##\text{ker} \ (A-I)=\text{ker} \ (I-A)=\text{im} \ (A)=U'##. Also...
13. ### I Linear Transformations: Why w1 is a Linear Combination of v

Given w = T (v), where T is a linear transformation and w and v are vectors, why is it that we can write any coefficient of w, such as w1 as a linear combination of the coefficients of v? i.e. w1 = av1 + bv2 + cv3 Supposably this is a consequence of the definition of linear transformations, but...
14. ### MHB Linear Transformations: Proving Rules & Demonstration

Good afternoon people. So i have to demonstrate that the problems below are Linear Transformations, i have searched and i know i have to do it using a couple of "rules", it is a linear transformation if: T(u+v) = T(u) + T(v) and T(Lu) = LT(u), the thing is that i really can't understand how to...
15. ### MHB -307.17.1 Show that S and T are both linear transformations

ok this is a clip from my overleaf homework reviewing just seeing if I am going in the right direction with this their was an example to follow but it also was a very different problem much mahalo
16. ### MHB Matrices of Linear Transformations .... Poole, Example 6.76 ....

I am reading David Poole's book: "Linear Algebra: A Modern Introduction" (Third Edition) and am currently focused on Section 6.6: The Matrix of a Linear Transformation ... ... I need some help in order to fully understand Example 6.76 ... ... Example 6.76 reads as follows: My question or...
17. ### MHB Combination of Linear Transformations

Hello, I'm trying to get my head around linear transformations, and there are a few things I'm not grasping too well. I'm trying to understand combinations of linear transformations, but I can't find a lot of clear information on them. As far as I can tell, any two linear transformations of the...
18. ### [Linear Algebra] Help with Linear Transformations part 2

Homework Statement Homework Statement (a) Let ##V## be an ##\mathbb R##-vector space and ##j : V \rightarrow V## a linear transformation such that ##j \circ j = id_V##. Now, let ##S = \{v \in V : j(v) = v\}## and ##A = \{v \in V : j(v) = -v\}## Prove that ##S## and ##A## are subspaces and...
19. ### [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)##...
20. ### [Linear Algebra] Linear Transformations, Kernels and Ranges

Homework Statement Prove whether or not the following linear transformations are, in fact, linear. Find their kernel and range. a) ## T : ℝ → ℝ^2, T(x) = (x,x)## b) ##T : ℝ^3 → ℝ^2, T(x,y,z) = (y-x,z+y)## c) ##T : ℝ^3 → ℝ^3, T(x,y,z) = (x^2, x, z-x) ## d) ## T: C[a,b] → ℝ, T(f) = f(a)## e) ##...
21. ### I Solutions to equations involving linear transformations

I have learned that for matrix theory, for ##A \vec{x} = \vec{b}##, if there exists a particular solution ##p##, then every solution looks like ##p+k##, where ##k \in \ker A##. Can someone help me find material on this online, but only in the context of general linear transformations? For...
22. ### Proving a statement about the rank of transformations

Homework Statement How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##? Homework Equations Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...
23. ### MHB The "Operator Norm" for Linear Transformations .... Browder, Lemma 8.4, Section 8.1, Ch. 8 .... ....

The "Operator Norm" for Linear Transformations ... Browder, Lemma 8.4, Section 8.1, Ch. 8 ... ... I am reader 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...

32. ### I Eigen Vectors, Geometric Multiplicities and more....

My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the...
33. ### MHB Yet Another Basic Question on Linear Transformations and Their Matrices

I am revising the basics of linear transformations and trying to get a thorough understanding of linear transformations and their matrices ... ... At present I am working through examples and exercises in Seymour Lipshutz' book: Linear Algebra, Fourth Edition (Schaum Series) ... ... At...
34. ### MHB (Very) Basic Questions on Linear Transformations and Their Matrices

Firstly, my apologies to Deveno in the event that he has already answered these questions in a previous post ... Now ... Suppose we have a linear transformation T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2 , say ... Suppose also that \mathbb{R}^3 has basis B and \mathbb{R}^2 has basis B'...
35. ### MHB Matrices of Linear Transformations .... Example 2.6.4 - McInerney

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.6 Constructing Linear Transformations ... I need help with a basic...
36. ### MHB Vector Spaces and Linear Transformations - Cooperstein Theorem 2.7

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 2.1 Introduction to Linear Transformations ... ... I need help with understanding Theorem 2.7 ... Theorem 2.7, its proof and some remarks read as follows:I am having considerable trouble...
37. ### Insights Matrix Representations of Linear Transformations - Comments

Fredrik submitted a new PF Insights post Matrix Representations of Linear Transformations Continue reading the Original PF Insights Post.
38. ### How Does the Linear Operator $$\phi$$ Transform Matrices to Polynomials?

Homework Statement Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where B= \begin{bmatrix} 3 & -2 \\ 2 & -2 \\ \end{bmatrix} Find rank,defect and one basis of an image and kernel of linear operator...
39. ### Linear Transformations p2 to R2

Homework Statement f: p2 to R2, f(ax2+bx+c) = (a+b, b+c) = V12. Homework Equations Create a p2 to R2 polynom and R2 equation that is equivalent to the above statement, so: f(dx2+ex+g) = (d+e, e+g) = V2 Therefore a=d b=e c=g f(ax2+bx+c) = (a+b, b+c) = f(dx2+ex+g) = (d+e, e+g) The Attempt at...
40. ### Onto equivalent to one-to-one in linear transformations

Can't quite see why a one-to-one linear transformation is also onto, anyone?
41. ### Linear transformation 2 x 2 matrix problem

Homework Statement [/B] Find a 2 x 2 matrix that maps e1 to –e2 and e2 to e1+3e2Homework Equations [/B] See the above notesThe Attempt at a Solution [/B] I am making a pig's ear out of this one. I think I can get e1 to –e2 3 -1 1 -3 but as far as getting it to reconcile a matrix like...
42. ### Linear Transformations and matrix representation

Assume the mapping T: P2 -> P2 defined by: T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2 is linear.Find the matrix representation of T relative to the basis B = {1,t,t2} My book says to first compute the images of the basis vector. This is the point where I'm stuck at because I'm not...
43. ### Derivatives and Linear transformations

Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer. I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?
44. ### Linear Transformations, Linear Algebra Question

Hi can anyone give me some hints with this question thanks A = \begin{pmatrix} 3 & -2 &1 & 0 \\ 1 & 6 & 2 & 1 \\ -3 & 0 & 7 & 1 \end{pmatrix} be a matrix for T:ℝ4→ℝ3 relative to the basis B = {v1, v2, v3, v4} and B'= {w1, w2, w3} v1 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 1 \end{pmatrix} v2 =...
45. ### Vector space, linear transformations & subspaces

Homework Statement Let V be a vector space over a field F and let L and M be two linear transformations from V to V. Show that the subset W := {x in V : L(x) = M(x)} is a subspace of V .The Attempt at a Solution I presume it's a simple question, but it's one of those where you just don't...
46. ### One-to-One Linear Transformations

Homework Statement Give a thorough explanation as to why a linear transformation: with a standard matrix A CANNOT be one to one. Homework Equations The Attempt at a Solution I think I have figured this one out, but I was hoping somebody could confirm whether this example is sufficient...
47. ### Linear Transformations and Image of a Matrix

Homework Statement Consider a 2x2 matrix A with A2=A. If vector w is in the image of A, what is the relationship between w and Aw? Homework Equations Linear transformation T(x)=Ax Image of a matrix is the span of its column vectors The Attempt at a Solution I know that vector w is one of the...
48. ### MHB Solving Linear Transformations with Matrix A and Vectors u1 & u2

This is the new topic we are going over in my Linear Algebra class and I am completely lost in how it works. From what I gather from the book, the lecture, and other sources amongst the internet, there are supposed to be a rule set shard with the transformation. The problem being is I cannot...
49. ### Linear transformations: function arguments

I have a small confusion about functions and variables. So, on doing a bit of reading, a linear transformation is a function that maps inputs from one vector space to another. So, let us take for example a simple rotation matrix. This matrix takes a point in 2D space and maps it to another...
50. ### Question about linear transformations

Homework Statement Hey PF, I'm here again asking about linear transformations, ha ha. Let C={(x,y) \in \mathbb R2 | x²+y²≤1} a circle of radius 1 and consider the linear transform T:\mathbb R2→\mathbb R2 (x,y) \mapsto (\frac{5x+3y}{4},\frac{3x+5y}{4}) Find all values of a natural n for...