Matrix Transformations from R^n to R^n

In summary, the conversation discusses the rotation matrix \(A(\theta)\) that rotates vectors in \(\mathbb{R}^2\) by an angle of \(\theta\). It is written as a 2x2 matrix and its effect on the standard basis is illustrated. The first column of the matrix represents the effect of (1,0) after rotation, while the second column represents the effect of (0,1). To find the values, the angle-sum identities are used.
  • #1
Swati
16
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.
 
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  • #2
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
 
  • #3
Sorry, I'm not getting it. Can you explain in brief.
 
  • #4
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
 
  • #5
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.)
 
  • #6
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).
 

FAQ: Matrix Transformations from R^n to R^n

What is a matrix transformation from R^n to R^n?

A matrix transformation is a mathematical operation that takes a vector from one coordinate system to another. In the context of R^n to R^n, it means transforming a vector from one n-dimensional space to another n-dimensional space using a matrix as the transformation tool.

How is a matrix transformation represented?

A matrix transformation can be represented by a matrix with the same dimensions as the original vector. Each element in the matrix represents how the original vector's corresponding element will be transformed.

What is the purpose of matrix transformations from R^n to R^n?

Matrix transformations are useful for a variety of applications in mathematics, physics, computer graphics, and other fields. They allow us to manipulate and transform vectors in different coordinate systems, making complex calculations and visualizations easier.

What are some common types of matrix transformations from R^n to R^n?

Some common types of matrix transformations from R^n to R^n include scaling, rotation, reflection, shearing, and projection. These transformations can be applied individually or in combination to achieve different effects on the original vector.

How do I perform a matrix transformation from R^n to R^n?

To perform a matrix transformation from R^n to R^n, you need to multiply the original vector by the transformation matrix. The resulting vector will be the transformed version of the original vector in the new coordinate system. It is important to note that the order of multiplication matters, and different transformation matrices will result in different transformations.

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