In this worksheet, we will practice finding the matrix of a linear transformation which rotates every vector in R² by a given angle.

**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: **

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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 centre 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?

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- 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 , which rotates each vector 90 degrees counterclockwise about the positive -axis then 45 degrees counterclockwise about the positive -axis. Find the matrix representation of in the standard basis.

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**Q16: **

Consider the linear transformation , which rotates each vector 45 degrees counterclockwise about the positive -axis then 90 degrees counterclockwise about the positive -axis. Find in the standard basis.

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