# Worksheet: Linear Transformations in Planes: Rotation

In this worksheet, we will practice finding the matrix of linear transformation of rotation at a given angle and the image of a vector under a given rotation linear transformation.

Q1:

Find the matrix for the linear transformation that rotates every vector in through an angle of .

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Q2:

Find the matrix for the linear transformation that rotates every vector in through an angle of .

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Q3:

Find the matrix for the linear transformation which rotates every vector in through an angle of .

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Q4:

Find the matrix for the linear transformation which rotates every vector in through an angle of .

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Q5:

Find the matrix for the linear transformation which rotates every vector in through an angle of .

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Q6:

Find the matrix for the linear transformation which rotates every vector in through an angle of .

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Q7:

Find, with respect to the standard basis, the matrix which rotates every vector in counterclockwise about the origin through an angle of .

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Q8:

Find, with respect to the standard basis, the matrix which rotates every vector in counterclockwise about the origin through an angle of .

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Q9:

Find, with respect to the standard basis, the matrix which rotates every vector in counterclockwise about the origin through an angle of .

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Q10:

Find, with respect to the standard basis, the matrix which rotates every vector in counterclockwise about the origin through an angle of .

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Q11:

Describe the geometric effect of the transformation produced by the matrix .

• Aa rotation through an angle of
• Ba rotation through an angle of
• Ca rotation through an angle of
• Da rotation through an angle of
• Ea rotation through an angle of

Q12:

A rotation with center the origin sends the vector to . Find the matrix representation of this rotation.

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Q13:

Let be the matrix of the transformation which rotates all vectors in through an angle of , where . For which values of does have a real eigenvalue?

• A and
• B and
• C and
• D and
• E and

Q14:

Consider the linear map , which acts by rotating a vector counterclockwise around the origin. Which of the following is the matrix of the linear transformation?

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Q15:

Consider the linear transformation , where rotates each vector 90 degrees about the positive -axis. Find the matrix representation of in the standard basis.

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Q16:

A linear transformation of a plane sends the vector to . If the transformation is a rotation about the origin, where does it send ?

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Q17:

A linear transformation of a plane sends vector to . If the transformation is a rotation, where does it send ?

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