MHB Matrix Transformations from R^n to R^n

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Multiplication by the rotation matrix \(A(\theta)\) rotates a vector \(X\) in the xy-plane by an angle \(\theta\). The transpose of this matrix, \(A^T\), effectively performs a rotation in the opposite direction, rotating the vector by \(-\theta\). The standard rotation matrix for \(\mathbb{R}^2\) is given by \(A(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}\). When applying this to the standard basis vectors, the first column corresponds to the image of (1,0) after rotation, while the second column corresponds to the image of (0,1). Understanding these transformations is crucial for grasping matrix operations in linear algebra.
Swati
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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.
 
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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 'How to define a vector field?'

Thread 'How to define a vector field?'

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