Linear Algebra - Matrix Transformations

In summary, the conversation discusses the composition of a rotation operator followed by a reflection operator in R2. The reflection operator is a matrix transformation with a standard matrix that reflects points about a line, and the rotation operator is also a matrix transformation with a standard matrix. The goal is to show that the composition of these two operators is another reflection operator. The solution involves using matrix multiplication and simplifying using trigonometric identities.
  • #1
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Homework Statement



Let L denote the line through the origin in R2 that that makes angle -∏ < theta ≤ ∏
with the positive x-axis. The reflection operator that reflects points about L in R2
is the matrix transformation R2 --> R2 with standard matrix

[cos 2(theta) sin 2(theta); sin 2(theta) -cos 2(theta)]

Show that the composition of a rotation operator followed by a re
reflection operator is another reflection operator.

Homework Equations



Standard matrix for reflection in R2 comes to mind;

[cos(theta) -sin(theta); sin (theta) cos (theta)]

These are 2x2 matrices and I'm not sure how to input them here so I put ; to separate the two columns.

The Attempt at a Solution



I'm kind of confused as to how to attempt to solve the question. I mean I could use the 2 standard matrices and use matrix multiplication which gives a really ugly 2x2 that I'm not sure what to do with. Any help to get me started would be appreciated :).
 
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  • #2
Anybody? : (
 
  • #3
It's pretty straight forward isn't it? You know how to write the rotation and reflection as matrices. The "rotation followed by a reflection" is the product of those two matrices. Do that multiplication and show that it is a reflection by determining the appropriate angle for the "line of reflection"?
 
  • #4
Thanks HallsofIvy! It turns out that I read the question wrong. I just had to use matrix multiplication and simplify. When I did it yesterday, I wasn't sure about how to simplify it. But using trig identities makes it very simple. Thanks again for your help!
 

FAQ: Linear Algebra - Matrix Transformations

What is a matrix transformation?

A matrix transformation is a mathematical operation in which a matrix is applied to a vector in order to transform it into a new vector. This transformation can involve scaling, rotating, shearing, or reflecting the vector, among other possibilities.

What is the purpose of matrix transformations?

Matrix transformations are used in linear algebra to represent and manipulate geometric transformations, such as rotations, translations, and reflections, in a convenient way. They are also important in computer graphics, physics, and other fields where transformations are needed.

How do you perform a matrix transformation?

To perform a matrix transformation, you first need to have a matrix representing the desired transformation. Then, you multiply this matrix by a vector representing the point or object you want to transform. The resulting vector will be the transformed version of the original vector.

Can matrix transformations be represented visually?

Yes, matrix transformations can be represented visually. For example, a 2D transformation can be represented as a transformation on a 2D coordinate plane, where the x and y axes are transformed according to the matrix. Similarly, a 3D transformation can be represented as a transformation on a 3D coordinate system.

Are there any limitations to matrix transformations?

Yes, there are some limitations to matrix transformations. One limitation is that they cannot perform non-linear transformations, such as curves or bends. Another limitation is that not all transformations can be represented as a matrix transformation. For example, a scaling transformation in which the x and y coordinates are scaled by different amounts cannot be represented by a single matrix.

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