Linear Algebra: Solving Rotation by Angle θ

In summary, The linear map fθ : R2 → R2 describes the rotation by the angle θ in the counterclockwise direction. To find the map, we need to find the values of a, b, c, and d such that fθ(x1, x2) = (ax1+bx2, cx1+dx2). Using the given equations, we can find that cosθ = ax1 + bx2 and sinθ = cx1 + dx2. By setting sinθ = icx1 + idx2 and cosθ + isinθ = e^(iθ), we can solve for a, b, c, and d. Additionally, to find the values of a and c,
  • #1
theshonen8899
10
0
This is for my Linear Algebra class:

Homework Statement



For an angle θ ∈ [0, 2π), find the linear map fθ : R2 → R2, which describes the rotation
by the angle θ in the counterclockwise direction.

Hint : For a given angle θ, find a, b, c, d ∈ R such that fθ(x1, x2) = (ax1+bx2, cx1+dx2).

Homework Equations



e^(x+yi) = (e^x)*(cos(y) + sin(y)i)

The Attempt at a Solution



Circle in counterclockwise direction is (cosθ, sinθ)
therefore
cosθ = ax1 + bx2
sinθ = cx1 + dx2
i(sinθ = cx1 + dx2) = isinθ = icx1 + idx2

cosθ + isinθ = ax1 + bx2 + icx1 + idx2 = e^(0 + iθ) = e^(iθ)

This isn't much but I've really been working on this problem the entire day and I really have no clue what I'm supposed to do. I feel like a damned fool for having to resort to this but I'd really like to have a solution to this before I head off to my quiz.

Thanks guys.
 
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  • #2
Hint: Calculate fθ(1,0) and fθ(0,1).
 

Related to Linear Algebra: Solving Rotation by Angle θ

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations, vector spaces, and linear transformations. It involves the use of algebraic techniques to solve problems related to lines, planes, and higher-dimensional objects. Linear algebra is widely used in many fields, including physics, engineering, computer science, and economics.

2. What is a rotation by angle θ?

A rotation by angle θ is a transformation that rotates a point or an object around a fixed point by a certain angle θ. This rotation can be clockwise or counterclockwise, depending on the sign of θ. In linear algebra, rotations are often represented by matrices and can be performed in higher dimensional spaces.

3. How is linear algebra used to solve rotations by angle θ?

In linear algebra, rotations by angle θ can be solved using rotation matrices. These matrices are square matrices that represent the rotation transformation in a specific dimension. By multiplying the rotation matrix by a vector representing a point, the resulting vector will be the rotated point. This process can be repeated for multiple points to rotate an entire object.

4. What is the difference between a rotation matrix and a transformation matrix?

A rotation matrix is a specific type of transformation matrix that is used to perform rotations. It has a specific form that simplifies the process of rotating objects. On the other hand, a transformation matrix is a more general matrix that can represent various types of transformations, such as translations, reflections, and rotations.

5. How are rotations by angle θ used in real-world applications?

Rotations by angle θ have numerous applications in fields such as computer graphics, robotics, and physics. In computer graphics, rotations are used to create 3D animations and simulate realistic movements. In robotics, rotations are necessary to control the movement of robotic arms and joints. In physics, rotations are used to describe the motion of objects and calculate forces and torques.

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