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Matrix Transformations from R^n to R^n

Swati

New member
Oct 30, 2012
16
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.
 

CaptainBlack

Well-known member
Jan 26, 2012
890
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
 

Swati

New member
Oct 30, 2012
16
Sorry, I'm not getting it. Can you explain in brief.
 

CaptainBlack

Well-known member
Jan 26, 2012
890
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
 

Swati

New member
Oct 30, 2012
16
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.)
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
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).