In this worksheet, we will practice finding the matrix of a linear transformation and describing it geometrically.

**Q1: **

Consider the linear transformation which maps to and to .

Find the matrix which represents this transformation.

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Where does this transformation map and ?

- A ,
- B ,
- C ,
- D ,
- E ,

**Q2: **

Suppose the linear transformation sends to and to . What is the absolute value of the determinant of the matrix representing ?

**Q3: **

A square of area 1 undergoes a linear transformation. Given that the area of the image is also 1, what can you say about the matrix of the transformation?

- AIt preserves angles.
- BIt is either a rotation or a reflection.
- CIt preserves distances.
- DIt has determinant 1 or .

**Q4: **

The determinant of a matrix is . What is the area of the image of a unit square under the transformation it represents?

**Q5: **

Consider the linear transformations for which , the image of , and , the image of , are unit vectors. Let be a linear transformation of this kind which has the additional property that the area of the parallelogram with vertices , , , and is as big as possible. What are the possible values of the size of the angle between and for the transformation ?

- A only
- B only
- C
- D and
- E

**Q6: **

A linear transformation maps the points , , , and onto , , , and , as shown.

By finding the areas of the object and image, and considering orientation, find the determinant of the matrix representing this transformation.

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**Q7: **

Find the matrix of the transformation that maps the points , , , and onto , , , and as shown.

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**Q8: **

What is the matrix that sends points , , and to , , and as shown?

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**Q9: **

A linear transformation is formed by rotating every vector in through an angle of and then reflecting the resulting vector in the -axis. Find the matrix of this linear transformation.

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**Q10: **

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

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**Q11: **

The matrix represents a linear transformation which sends the vector to . What can you say about the matrix ?

- AIts first row is .
- BIts second column is .
- CIts second row is .
- DIts first column is .

**Q12: **

Suppose that the matrix represents a transformation that sends the vector to itself and sends every vector in the -plane to a (possibly different) vector in the -plane. What can be said about the entries of ?

- A , , , ,
- B , , , ,
- C , , , ,
- D , , , ,
- E , , , ,

**Q13: **

Suppose that the matrix represents a transformation that sends the vector to itself and sends every vector in the -plane to a (possibly different) vector in the -plane. What can be said about the entries of ?

- A , , , ,
- B , , , ,
- C , , , ,
- D , , , ,
- E , , , ,

**Q14: **

In two-dimensional space you only need to specify the angle of a rotation. In three-dimensional space you need to give both an angle and a vector which represents the axis of rotation. Consider the matrix which represents a rotation of by about an axis through the origin and in the direction of

What is ?

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Find the general form of the matrix which sends to the appropriate vector as determined in the previous part of the question.

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The vector is perpendicular to . What can you say about the direction of the vector ?

- A will be perpendicular to but not necessarily to
- B will be parallel to
- C will be perpendicular to both and
- D will be parallel to
- E will be perpendicular to but not necessarily to

What can you say about the magnitude of ?

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Which of the following vectors has the required properties to be ?

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What can you say about the vector ?

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Using the general form of the matrix from the second part of the question, and the values of and , find the matrix .

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