Transformation matrix Definition and 68 Threads

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...

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...

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...

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...

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...

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...

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...

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...

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...

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}...

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

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...

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...

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?

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...

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...

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...

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.