What is Transformation matrix: Definition and 69 Discussions

In linear algebra, linear transformations can be represented by matrices. If


{\displaystyle T}
is a linear transformation mapping



{\displaystyle \mathbb {R} ^{n}}



{\displaystyle \mathbb {R} ^{m}}


{\displaystyle \mathbf {x} }
is a column vector with


{\displaystyle n}
entries, then





{\displaystyle T(\mathbf {x} )=A\mathbf {x} }
for some


{\displaystyle m\times n}


{\displaystyle A}
, called the transformation matrix of


{\displaystyle T}
. Note that


{\displaystyle A}


{\displaystyle m}
rows and


{\displaystyle n}
columns, whereas the transformation


{\displaystyle T}
is from



{\displaystyle \mathbb {R} ^{n}}



{\displaystyle \mathbb {R} ^{m}}
. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.

View More On Wikipedia.org
  1. V

    Transformations to both sides of a matrix equation

    I feel if we have the matrix equation X = AB, where X,A and B are matrices of the same order, then if we apply an elementary row operation to X on LHS, then we must apply the same elementary row operation to the matrix C = AB on the RHS and this makes sense to me. But the book says, that we...
  2. J

    Transformation Matrix T in Terms of β1, β2 with Row Reduction Explained

    T(α1), T(α2), T(α3) written in terms of β1, β2: Tα1 =(1,−3) Tα2 =(2,1) Tα3 =(1,0). Then there is row reduction: Therefore, the matrix of T relative to the pair B, B' is I don't understand why the row reduction takes place? Also, how do these steps relate to ## B = S^{-1}AS ##? Thank you.
  3. patric44

    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...
  4. F

    I Change of Basis Matrix vs Transformation matrix in the same basis....

    Hello, Let's consider a vector ##X## in 2D with its two components ##(x_1 , x_2)_A## expressed in the basis ##A##. A basis is a set of two independent (unit or not) vectors. Any vector in the 2D space can be expressed as a linear combination of the two basis vectors in the chosen basis. There...
  5. johnconner

    I Transformation matrix for an expanding space

    Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this: \begin{pmatrix}...
  6. R

    I Beam-splitter transformation matrix

    The transformation matrix for a beam splitter relates the four E-fields involved as follows: $$ \left(\begin{array}{c} E_{1}\\ E_{2} \end{array}\right)=\left(\begin{array}{cc} T & R\\ R & T \end{array}\right)\left(\begin{array}{c} E_{3}\\ E_{4} \end{array}\right) \tag{1}$$ Here, the amplitude...
  7. D

    I Index placement -- Lorentz transformation matrix

    Hi. I came across the following statement , which seems wrong to me. Λμρ = ( ΛT )ρμ I have it on good authority (a previous post on this forum) that (ΛT)μν = Λνμ so I am hoping that the first equation is wrong ? It looks like the inverse not the transpose ? The equation Λμρ η μνΛνσ = ηρσ is...
  8. C

    MHB Finding the Rotation Matrix for 60 Degree Rotation around (1,1,1) Axis

    Dear Everybody, I am having some problem with one exercise. And the question states: Find the transformation Matrix R that describes a rotation by 60 degrees about an axis from the origin thru the pt (1,1,1). The rotation is clockwise as you look down toward the origin. I know the standard...
  9. Ben Geoffrey

    I Lorentz Transformation Matrix: Tensor of Order 2?

    Is the Lorentz transformation matrix Λμν a tensor of order two and does it transform like a tensor ?
  10. Akineton

    I Transformation matrix from Dirac to Weyl

    Hello friends, I'm trying to construct transformation matrix S such that it transforms Dirac representations of gamma matrices into Chiral ones. I know that this S should be hermitian and unitary and from this I arrived an equation with 2 matrices on the LHS (a known matrix multiplied by S from...
  11. K

    A Is My Transformation Matrix Correct?

    Hi, I have attached a pdf which shows clearly how I have carried out my transformations from one axis into another. However, I am not convinced that it is right and I have described why I feel so. I shall be grateful if someone can help me Kajal
  12. G

    Row space of a transformation matrix

    Homework Statement We're given some linear transformations, and asked what the null space, column space and row space of the matrix representations tell us Homework EquationsThe Attempt at a Solution I know what information the column space and null space contain, but what does the row space of...
  13. F

    I Transformation of Tensor Components

    In the transformation of tensor components when changing the co-ordinate system, can someone explain the following: Firstly, what is the point in re-writing the indicial form (on the left) as aikTklajl? Since we're representing the components in a matrix, and the transformation matrix is also...
  14. R

    Finding Coordinate Matrix for Linear Transformation T

    Homework Statement Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question. Consider the...
  15. R

    Linear Algebra matrix linear transformation

    Homework Statement Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2 Let E = (e1, e2, e3) be the ordered basis in P2 given by e1(t) = 1, e2(t) = t, e3(t) = t^2 Find the coordinate matrix...
  16. fluidistic

    Length contraction via Lorentz transformation matrix

    1,2,3. Homework Statement I tried to derive the length contraction using the Lorentz transformation matrix and considering 2 events. I reached the correct result but there's a step that I had to assume that I don't understand. Consider a ruler of length L along the x-axis for an observer at...
  17. D

    Lorentz transformation matrix and its inverse

    Given the Lorentz matrix Λuv its transpose is Λvu but what is its transpose ? I have seen ΛuaΛub = δb a which implies an inverse. This seems to be some sort of swapping rows and columns but to get the inverse you also need to replace v with -v ? Also In the LT matrix is it the 1st slot...
  18. B

    How to form the transformation matrix for this

    We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis. The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or...
  19. Th3HoopMan

    QR Decomposition w/ Householder and Givens Transformations

    Could anybody link me to some good examples on how to go about doing them? I honestly have no idea how to go about doing these two types of problems.
  20. R

    Quantum Mechanics: Transformation Matrix

    Homework Statement Determine a ##2\times 2## matrix ##\mathbb{S}## that can be used to transform a column vector representing a photon polarization state using the linear polarization vectors ##|x\rangle## and ##|y\rangle## as a basis to one using the circular polarization vectors...
  21. J

    Function scales eigenvalues, but what happens to eigenvectors?

    Statement: I can prove that if I apply a function to my matrix (lets call it) "A"...whatever that function does on A, it will do the same thing to the eigenvalues (I can prove this with a similarity transformation I think), so long as the function is basically a linear combination of the powers...
  22. L

    How to Derive the Grasp Transformation Matrix for a Three-Finger Robot Hand?

    Hello everyone, i am now working on a problem with three fingers robot hand to grab a cube to undergo some motion however i face some difficulties on deriving the grasp transformation matrix which help to switching the local coordinate frame at first i was given three point vectors [0 1...
  23. PhizKid

    Transformation matrix with respect to two bases?

    Homework Statement Let ##S = \{1, e^x, e^{-x}, e^{2x}, e^{-2x}\}## and ##B = \{1, sinh(x),cosh(x), sinh(2x), cosh(2x)\}##. S spans the vector space V, and a linear transformation T: V -> V is defined by T(y) = y'' - 3y' - 4y. (a) Find the representation matrix of T with respect to the bases S...
  24. Petrus

    MHB Solving Equation System for Surjective Linear Transformation: T:R^4->R^2

    T is a surjective linear transformation T: \mathbb{R^4}-> \mathbb{R^2}. Decide dim ker T. How many free variables do I get if I solve equation system T(x)=y for a vector y \in \mathbb{R^2}? Construct a transformation matrix belonging to a surjective linear transformation...
  25. A

    Inverse transformation matrix entry bounds

    I have sets of 2d vectors to be transformed by an augmented matrix A that performs an affine transform. Matrix A can have values that differ at most |d| from the identity matrix, to limit the transformation, meaning that the min/max bounds for A are I_3 \pm dI_3 The problem is that i'd lke...
  26. Q

    Lorentz transformation matrix applied to EM field tensor

    In a recent course on special relativity the lecturer derives the Lorentz transformation matrix for the four vector of position and time. Then, apparently without proof, the same matrix is used to transform the EM field tensor to the tensor for the new inertial frame. I am unclear whether it...
  27. U

    Show that the linear transformation matrix is a contraction mapping

    Homework Statement Show that the following linear transformation matrix is a contraction mapping. \begin{bmatrix} 0.5 & 0 & -1 \\ 0 & 0.5 & 1 \\ 0 & 0 & 1 \end{bmatrix} I don't know how to make that into a matrix, but it is a 3x3 matrix. The first row is [.5 0 -1] the second row is [0...
  28. X

    Transformation matrix of linear n-dimensional state-space system

    Hi all, I have a linear algebra question relating actually to control systems (applied differential equations) for the linear system {\dot{\vec{{x}}} = {\bf{A}}{\vec{{x}}} + {\bf{B}}}{\vec{{u}}}\\ \\ A \in \mathbb{R}^{ nxn }\\ B \in \mathbb{R}^{ nx1 }\\ In class, we formed a...
  29. M

    The Lorentz transformation matrix properties

    Hello, As known, any Lorentz transformation matrix \Lambda must obey the relation \Lambda^\mu~_\nu\Lambda^\rho~_\sigma g_{\mu \rho}=g_{\nu \sigma}. The same holds also for the inverse metric tensor g^{\nu \sigma} which has the same components as the metric tensor itself (don't really...
  30. Philosophaie

    Transformation Matrix from x-axis

    I have a Parametric Equation for a Cone: x=u y=cos(v)*a*(u-h)/h z=sin(v)*a*(u-h)/h where: h is height of the cone a is the Radius of the Base u goes from 0 to h v goes from 0 to 2*pi This cone lies on the x-axis. I need it to lie on the theta and phi axis. This is what I came up with to...
  31. N

    Transformation matrix from global to loca frame

    Hello friends Iam naveen and new to this forum.I have a question regarding the transformation matrix from world frame to the base frame of a Spatial serial manipulator. The frames are attached like this. The Z-axes are along the link lengths. The Xi axis is oriented such that Xi = Zi X Zi+1...
  32. C

    Can't understand technique for computing transformation matrix

    My linear algebra textbook presents a technique for computing the transition matrix from an old basis B' to a new basis B. Apparently if you set up an augmented matrix with B on the left and B' on the right, then put B into RREF, the resulting matrix on the right after those row operations will...
  33. A

    Resetting the Affine Transformation matrix

    Affine Transformation Matrix is said to be formed by initializing it using a learned projection matrix from a conventional algorithm like Eigenfaces or Fisherfaces; then it is reset by using the singular value decomposition T=UAV', where T is the transformation matrix. Could somebody explain...
  34. W

    Transformation matrix for components of acceleration

    Hello any body can find my mistake ?! TO find the component of a vector in other coordinate we can use the transformation matrix : http://up98.org/upload/server1/01/z/ff96m5hl2uahgjw3u2un.jpg but why this does nt work for acceleration vector ? i mean why i can't derive the component of...
  35. D

    Find the linear transformation matrix

    Homework Statement Suppose that a linear transformation maps a point (2,3) to (0,1) and maps a point (9,7) to (1,0). Find the matrix for this linear transformation. 2. Solution (answersheet) Observe that the two point that are the result of the mapping are the two base vectors. If our...
  36. G

    Deriving transformation matrix from clues

    I have PTV=R where P and T are square matrices (4x4) and V and R are non-square (4x3). P and V are known, T is unknown, and R is partially known (3 unknown elements). Seems impossible, but T is a transformation matrix (ie upperleft 3x3 is a rotation matrix) which gives me additional clues...
  37. R

    Sensor Body to ENU transformation Matrix

    Hi, I want to convert the accelerometer values from sensor frame to ENU reference frame , I am using oreintation sensor values, yaw ,pitch ,roll. I want to confirm one thing that After transforming to ENU frame should I get accelertation values with reference to ENU frame like this [0 0 Z]...
  38. T

    Extracting yaw, pitch, roll from transformation matrix

    There are two references frames, A and B. Let A's reference frame be denoted by the columns of the identity matrix, and let A's origin be (0,0,0). Let B's reference frame and origin be denoted by a transformation matrix T, where T = R11 R12 R13 x R21 R22 R23 y R31 R32 R33 z 0 0 0 1...
  39. L

    D'alembertian of lorentz transformation matrix

    Is the d'alembertian of lorentz transformation matrix 0? and why? would it be 0 because it lorentz invariant? thanks
  40. P

    Exploring Lorentz Transformation Matrix: A'^{\mu}=\alpha^{\mu}_{\nu}A^{\nu}

    \alpha=\left(\begin{array}{cccc} \gamma& 0&0& -\beta\gamma\\ 0&1& 0 & 0\\ 0 & 0 & 1 & 0\\ -\beta\gamma & 0 & 0 & \gamma \end{array} \right)x'^{\mu}=\alpha^{\mu}_{\nu} x^{\nu} \alpha is Lorrentz transformation matrix. Can I see something more about it? . It's symmetric. That is important...
  41. silvermane

    Finding the Transformation Matrix if it's linear

    Homework Statement Which of the transformations are linear? If they are, then find the transformation matrix. the input is v = (v1,v2) a. t(v) = (v2,v1) b. t(v) = (v1,v2) c. t(v) = (0,v1) d. t(v) = (0,1) The Attempt at a Solution a. it is linear b. it is linear c. I think it is...
  42. silvermane

    Finding a Transformation Matrix to yield the basis

    Homework Statement The solutions to the linear differential equation d^2u/dt^2 = u for a vector space. Find two independent solutions, to give a basis for that solution space. The Attempt at a Solution I want to understand this question. I feel that there's something I'm missing. I...
  43. B

    How Does Rapidity Influence the Lorentz Transformation Matrix?

    1. Homework Statement : Consider a two dimensional Minkowski space (1 spatial, 1 time dimension). What is the condition on a transformation matrix \Lambda, such that the inner product is preserved? Solve this condition in terms of the rapidity. 2. Homework Equations : Rapidity Relations...
  44. S

    Can a Transformation Matrix be one-to-one and not onto?

    Title says all.. A transformation matrix is one to one if its columns are linearly independant, meaning it has a pivot in each column but what if it doesn't have a pivot in each row(i.e. not onto)? is it still one-to one?
  45. S

    Finding the Image of a Vector under a Linear Transformation

    Homework Statement Let L: R^3 -> R^3 be a linear transformation such that L(i) = [1 2 -1], L(j) = [1 0 2] and L(k) = [1 1 3]. Find L([ 2 -1 3)]. All the numbers in [ ] should be vertical, but I don't know how to set that up. Homework Equations The Attempt at a Solution...
  46. Saladsamurai

    A Transformation Matrix question

    Homework Statement Find the transformation matrix 'R' that describes a rotation by 120 degrees about an axis from the origin through the point (1,1,1). The rotation is clockwise as you look down the axis toward the origin. Homework Equations \left( \begin{array}{c} A'_x \\ A'_y \\...
  47. C

    Linear Transformation matrix help

    The problem is as follows: Find a nonzero 2x2 matrix A such that Ax is parallel to the vector [1] [2] for all x in R2. So far I know A=[v1 v2] therefore Ax= [v1 v2][x1] [x2] = x1v1+x2v2 I know these two vectors are parallel, but I am a...
  48. P

    Retrieving angle of rotation from transformation matrix

    Hi! How do I calculate the angle of rotation for each axis by a given 4x4 transformation matrix? The thing is that all values are a kind of mixed up in the matrix, so I cannot get discrete values to start calculating with anymore. Thanks, Phong
  49. M

    Trying to derive a transformation Matrix from a set of known points

    Hi, I don't know if this should go in a Math forum or a Programming forums, but y'all here seem quite handy with mathematics, so I'll give it a shot. If this is totally not what y'all are about, just let me know. I have two computer images... one of them is an "original" image. The other one...
  50. B

    Compute transformation matrix in nth dimension

    I have a vector of all ones in n-dimensions. For example (1,1,1) in 3D. I want to find a invertible rotation matrix T that transforms the vector of all ones to the vector (0,0,0,...,0,,1): Let v be the vector of all ones, and w=(0,0,...,0,1) Find T such that T.v == wIn low dimension it is easy...