Finding the matrix of a transformation

In summary: If you don't know the answer to that (which is a sign that you might need to review some basic trigonometry), then think about this: what would be the coordinates of the points on the unit circle if you rotated them 45 degrees counterclockwise?
  • #1
Cankur
9
0

Homework Statement


Consider the transformation T from R2 to R2 that rotates any vector x through an angle of 45 degrees in the counterclockwise direction. You are told that T is a linear transformation. Find the matrix of T.


Homework Equations





The Attempt at a Solution



A vector with components x1 and x2 should become x 2 and x1, seeing that the transformation should rotate the vector 45 degrees. So therefore, the matrix of the transformation should be:

[0 1]
[1 0]
 
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  • #2
Cankur said:

Homework Statement


Consider the transformation T from R2 to R2 that rotates any vector x through an angle of 45 degrees in the counterclockwise direction. You are told that T is a linear transformation. Find the matrix of T.


Homework Equations





The Attempt at a Solution



A vector with components x1 and x2 should become x 2 and x1, seeing that the transformation should rotate the vector 45 degrees.
The rotation you're describing here is a rotation of 90° counterclockwise, not 45°.

What should T do to x1 = <1, 0> and x2 = <0, 1>?
Cankur said:
So therefore, the matrix of the transformation should be:

[0 1]
[1 0]
 
  • #3
Cankur said:

Homework Statement


Consider the transformation T from R2 to R2 that rotates any vector x through an angle of 45 degrees in the counterclockwise direction. You are told that T is a linear transformation. Find the matrix of T.


Homework Equations





The Attempt at a Solution



A vector with components x1 and x2 should become x 2 and x1, seeing that the transformation should rotate the vector 45 degrees. So therefore, the matrix of the transformation should be:

[0 1]
[1 0]
No, with a 45 degree rotation, any vector on the x-axis (y=0) would rotate to the line y= x while any vector on the y-axis would rotate to the line y= -x. What vectors on those lines have length 1?
 
  • #4
It should move it half of 90 degrees. But what matrix would achieve that? A matrix that looks like this perhaps:

[0 1/2]
[1/2 0]

How should you think when you have problems of this sort?
 
  • #5
Cankur said:
It should move it half of 90 degrees. But what matrix would achieve that? A matrix that looks like this perhaps:

[0 1/2]
[1/2 0]

How should you think when you have problems of this sort?
No, the matrix above doesn't work.

I'll ask this again.
Mark44 said:
What should T do to x1 = <1, 0> and x2 = <0, 1>?
 
  • #6
Which (because I just have to jump in behind Mark44) is much what I asked before: what vector, <x, x>, has length 1? What vector <-x, x>, has length 1?
 

1. What is a transformation matrix?

A transformation matrix is a square matrix that represents a linear transformation in a vector space. It consists of a set of numbers arranged in a grid-like pattern, and it is used to describe how a transformation affects the coordinates of a vector.

2. How is a transformation matrix calculated?

A transformation matrix is calculated by first determining the basis vectors of the input and output vector spaces. Then, the coordinates of each basis vector are used to form the columns of the transformation matrix. The resulting matrix represents the linear transformation from the input space to the output space.

3. What are the properties of a transformation matrix?

A transformation matrix has several properties, including being square (having the same number of rows and columns), having a determinant that is not equal to zero, and being invertible (having an inverse matrix). It also preserves the properties of linear transformations, such as preserving parallel lines and the origin.

4. How is a transformation matrix used in computer graphics?

In computer graphics, transformation matrices are used to represent the position, orientation, and scale of objects in a 3D space. By applying transformation matrices to the coordinates of each point in an object, it can be rotated, translated, and scaled in the virtual world.

5. Are there different types of transformation matrices?

Yes, there are different types of transformation matrices depending on the type of transformation being performed. Some common types include translation matrices, rotation matrices, scaling matrices, and shearing matrices. Each type has a specific form and purpose in representing different types of transformations.

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