Transformation matrix Definition and 69 Threads

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



T


{\displaystyle T}
is a linear transformation mapping





R


n




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





R


m




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




x



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



n


{\displaystyle n}
entries, then




T
(

x

)
=
A

x



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



m
×
n


{\displaystyle m\times n}
matrix



A


{\displaystyle A}
, called the transformation matrix of



T


{\displaystyle T}
. Note that



A


{\displaystyle A}
has



m


{\displaystyle m}
rows and



n


{\displaystyle n}
columns, whereas the transformation



T


{\displaystyle T}
is from





R


n




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





R


m




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

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  1. U

    Transformation Matrix for Derivative Operator Acting on Third Order Polynomials

    Homework Statement Find m(DT), that is, find the matrix for the transformation DT where D is the derivative operator and T: V -> V , T(p(x)) = xp'(x). The polynomial is of degree <= 3, and the basis for it is (1,x,x^2, x^3).Homework Equations Basic matrix multiplication needs to be...
  2. E

    Finding a basis given a transformation matrix

    Homework Statement Let T : M2,2, --> M2,1 be the linear transformation given by T ([a b; c d]) = [a-2b ; c-2d] Fix bases B = { [1 0 ; 0 0], [ 0 1 ; 0 0], [0 0 ; 1 0], [0 0 ; 0 1]} and C = { [1 ; 0], [0 ; 1]} for M2,2, and M2,1 respectively. (a) Find the matrix [T]C,B of T with...
  3. F

    Transformation matrix derivation problem

    Homework Statement As part of an assignment, I need to derive a transformation matrix to convert a vector in cartesian coordinates to spherical coordinates. Homework Equations What I've got so far is: For an arbitrary vector V, \textbf{V}=\left[\begin...
  4. K

    Transformation matrix on tensors

    Hello. I wasn't sure whether to post this here on in some of the physics sections. I have a rank 2 tensor in one coordinate reference system [x1, x2, x3], the one where only the principal elements are non zero: R=[ a11 0 0; 0 a22 0; 0 0 a33 ]. I want the tensor R in some other...
  5. G

    How Do Matrix Transformations Alter 3D Space Geometrically?

    Homework Statement 7. (a) A transformation, T1 of three dimensional space is given by r'=Mr, where r=\left( \begin{array}{c} x \\ y \\ z \end{array} \right) r'=\left( \begin{array}{c} x' \\ y' \\ z' \end{array} \right) and M=\left( \begin{array}{ccc} 1 & 0...
  6. B

    Finding the Matrix of a Linear Transformation

    Homework Statement Assume that T defines a linear transformation and use the given information to find the matrix of T T: R4-->R2 such that T(1,0,0,0)=(3,-2), T(1,1,0,0)=(5,1), T(1,1,1,0)=(-1,0), and T(1,1,1,1)=(2,2)Homework Equations The Attempt at a Solution I think I need to use/find the...
  7. W

    Is the linear transformation matrix T invertible

    Homework Statement T: M22 --> M22 defined by T(A) = AB where B = [ 3 2 ] [ 2 1 ] Is the linear transformation matrix T invertible with respect to the standard bases? If so, find it. Homework Equations none The Attempt at a Solution This is going to sound stupid, but I need...
  8. M

    Transformation matrix, vector algebra word problem

    Hi everyone. I am not sure if this problem belongs under the "Linear & Abstract algebra" section but it seemed like it may. Please let me know if there is a different section that would better fit this problem. So here is a word problem that is proposed: A solar panel is capable of rotating...
  9. chroot

    Finding an affine transformation matrix

    Hey guys, I have a problem in a computer vision application that requires me to find an affine transformation matrix, A. What I've got are four corners of a quadrilateral, in 2D coordinates on the image plane. These are the projections of the 3D corners of the real quadrilateral onto the...
  10. S

    Linear transformation matrix problem

    let A= \left( \begin{array}{Ccc} 9 & 0 \\ 2 & 6 \\ \end{array} \right) and B= \left( \begin{array}{Ccc} 5 & 1 \\ 3 & 4 \\ \end{array} \right) Find the matrix C of the linear transformation T(x)=B(A(x)). The Attempt at a Solution - Once again, I really don't know how to...
  11. A

    Co-ordinate transformation matrix

    Plz advise if my approach is correct for 1st part and for 2nd part, I need some help. Problem Statement Consider the linear transformation T: R3->R2 whose matrix with respect to standard bases is given by | 2,1,6 | | 0,2,-1|. Now consider the bases f1={2,4,0}...
  12. H

    Cartesian Tensors and transformation matrix

    I was just reading chapter on Cartesian tensors and came across equation for transformation matrix as function of basic vectors. I just do not get it and cannot find a derivation. I am too old to learn Latex, I uploaded a word document with the equation. Thanks, Howard
  13. S

    How Do You Calculate the Transformation Matrix for a Quadrilateral in 3D Space?

    Homework Statement Consider the quadrilateral (namely Q) in R^3 formed by the points (1, 0, 0), (2, 0, 0), (1, 1, 3), and (2, 1, 3). a) What should the coordinates be for the figure R we get by rotating Q counterclockwise in the x-y plane by 45 degrees, then dilating it by a factor of...
  14. D

    Transformation Matrix: Understanding Its Purpose and Properties

    Given a transformation matrix T, which maps objects (x,y) to the image (x',y'). The inverse of T will map the image back to the object. Just wondering, what happens if matrix T is singular i.e. det(T)=0? Then there is no matrix to map the images back to the object. My teacher said that...
  15. L

    Understanding Transformation Matrix Order for B and B'

    I was just wondering that when we take P, the transformation matrix from B to B', does B and B' have to be ordered from the highest thing? What I mean is that I have B = 1, 1+x, 3+4x+2x^2 When I do the actual transformation, must I order it and do 2x^2+4x+3 first?
  16. U

    How Do You Show |λ|^2 = 1 for a 2D Transformation Matrix?

    Show by direct expansion that | \lambda | ^2 =1 For simplicity, take \lambda to be a two-dimensional transformation matrix. from what I understand, if X was a vector (2,3,4), | X | is finding the length of the vector by adding the square of the numbers and taking a square root...
  17. U

    Find Transformation Matrix for 45° Rotation Around x_2 Axis

    Find the transformation matrix that rotates the axis x_3 of a rectangular coordinate system 45 degrees toward x_1 around the x_2 axis. I have a question about what exactly are the x_1,x_2,x_3 supposed to be. Do I assume that they are the x,y,z axis? Also, what is the general form of a...
  18. P

    How to Find the Transformation Matrix for a 120-Degree Rotation About (1,1,1)?

    I don't know if this is the right section, but this problem is in my electromagnetism course (Griffiths text). This is problem 1.9 of Griffiths (3rd edition) text: Find the transformation matrix R that describes a rotation by 120 degrees about an axis from the origin through the point...
  19. W

    Coordinate transformation matrix?

    Can anyone tell me: 1) How to understand the defination to orthogonal transformation matrix? Defination: A(i,j)A(k,j)=q(i,k) where q is Kronecker delta. 2) Why the inverse of this orthogonal matrix is equal to its transpose? Will.
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