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- Thread starter Swati
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- Jan 26, 2012

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Write out the matrix \(A(\theta)\) that rotates vectors by an angle \( \theta\). Now take its transpose, what do you notice?1. If multiplication by A rotates a vectorXin the xy-plane through an angle (theta). what is the effect of multiplyingxby A^T ? Explain Reason.

CB

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- Jan 26, 2012

- 890

What is the matrix \(A(\theta)\) (the rotation matrix that rotates vectors in \(\mathbb{R}^2\) by \(\theta\) ) written out in full?Sorry, I'm not getting it. Can you explain in brief.

CB

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A=[cos^2(theta)-sin^2(theta), -2sin(theta)cos(theta) ;What is the matrix \(A(\theta)\) (the rotation matrix that rotates vectors in \(\mathbb{R}^2\) by \(\theta\) ) written out in full?

CB

2sin(theta)cos(theta),

cos^2(theta)-sin^2(theta)]

(A is 2*2 matrix.)

- Feb 15, 2012

- 1,967

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).