Understanding R2 to R2 Linear Transformations

The video tutorial explains how to construct a standard transformation matrix for an R2 to R2 linear transformation that rotates points by 2/3 pi radians about the origin. It details the process of determining the transformation of basis vectors e1 and e2, calculating their images after rotation, and constructing the transformation matrix. The tutorial also compares the derived matrix with the general form of a rotation matrix in R2, verifying the results using trigonometric functions from the unit circle.

What is the purpose of a standard matrix in an R2 to R2 linear transformation?

• To translate vectors

• To rotate vectors

• To reflect vectors

• To scale vectors

How many radians does the transformation rotate points around the origin?

• 3/4 pi radians

• 2/3 pi radians

• Pi radians

• Pi/2 radians

What is the image of vector e1 (1,0) after a 120-degree counterclockwise rotation?

• (1/2, sqrt(3)/2)

• (sqrt(3)/2, 1/2)

• (-1/2, sqrt(3)/2)

• (-1/2, -sqrt(3)/2)

What is the transformation of vector e2 (0,1) after the rotation?

• (sqrt(3)/2, 1/2)

• (-sqrt(3)/2, -1/2)

• (-1/2, sqrt(3)/2)

• (1/2, -sqrt(3)/2)

What is the first column of the transformation matrix?

• (-1/2, sqrt(3)/2)

• (1/2, -sqrt(3)/2)

• (sqrt(3)/2, 1/2)

• (-1/2, -sqrt(3)/2)

What is the second column of the transformation matrix?

• (1/2, -sqrt(3)/2)

• (-1/2, sqrt(3)/2)

• (-sqrt(3)/2, -1/2)

• (sqrt(3)/2, 1/2)

What does the general form of a rotation matrix in R2 depend on?

• The reflection axis

• The translation vector

• The angle of rotation

• The scale factor

What is the cosine of 2/3 pi?

• -sqrt(3)/2

• -1/2

• 1/2

• sqrt(3)/2

What is the sine of 2/3 pi?

• 1/2

• -1/2

• -sqrt(3)/2

• sqrt(3)/2

What is the opposite of sine 2/3 pi?

• sqrt(3)/2

• -1/2

• 1/2

• -sqrt(3)/2