MHB Matrix Transformations from R^n to R^n

Swati
Messages
16
Reaction score
0
1. If multiplication by A rotates a vector X in the xy-plane through an angle (theta). what is the effect of multiplying x by A^T ? Explain Reason.
 
Physics news on Phys.org
Swati said:
1. If multiplication by A rotates a vector X in the xy-plane through an angle (theta). what is the effect of multiplying x by A^T ? Explain Reason.

Write out the matrix \(A(\theta)\) that rotates vectors by an angle \( \theta\). Now take its transpose, what do you notice?

CB
 
Sorry, I'm not getting it. Can you explain in brief.
 
Swati said:
Sorry, I'm not getting it. Can you explain in brief.

What is the matrix \(A(\theta)\) (the rotation matrix that rotates vectors in \(\mathbb{R}^2\) by \(\theta\) ) written out in full?

CB
 
CaptainBlack said:
What is the matrix \(A(\theta)\) (the rotation matrix that rotates vectors in \(\mathbb{R}^2\) by \(\theta\) ) written out in full?

CB

A=[cos^2(theta)-sin^2(theta), -2sin(theta)cos(theta) ;
2sin(theta)cos(theta),
cos^2(theta)-sin^2(theta)]

(A is 2*2 matrix.)
 
err...no, it's not.

suppose we rotate (counter-clockwise) through an angle of θ.

to get the matrix for such a rotation, we need to know its effect on a basis for the plane.

there's no compelling reason not to use the standard basis {(1,0),(0,1)}, so we will.

it should be (hopefully) obvious that after the rotation, (1,0) gets mapped to (cos(θ),sin(θ)). this tells you what the first column of the matrix should be (WHY?).

what does (0,1) get mapped to?

(HINT: 0 = cos(π/2), 1 = sin(π/2).

what is cos(π/2 + θ), sin(π/2 + θ)? use the angle-sum identities).
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Passive Transformation and Rotation Matrix
Replies
1
Views
3K
I Index notation of vector rotation
Replies
7
Views
2K
MHB Matrix Transforms: nxm, n->m, m->n, n+m->n/m
Replies
3
Views
2K
I Want to understand how to express the derivative as a matrix
Replies
8
Views
2K
I Matrix rank in terms of determinant polynomial
Replies
14
Views
3K
I Change of Basis Matrix vs Transformation matrix in the same basis....
Replies
12
Views
5K
I Orthogonal transformations preserve length
Replies
1
Views
2K
I Can one find a matrix that's 'unique' to a collection of eigenvectors?
Replies
33
Views
674
Vectors as geometric objects and vectors as any mathematical objects
Replies
27
Views
2K
MHB Diagonalizable transformation - Existence of basis
2
Replies
52
Views
3K

Hot Threads

Recent Insights

Back
Top