# Worksheet: Matrix of Linear Transformation

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

**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 has determinant 1 or .
- CIt is either a rotation or a reflection.
- DIt preserves distances.

**Q4: **

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

**Q5: **

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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**Q6: **

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

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**Q7: **

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

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**Q8: **

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

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

**Q9: **

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, , , ,

**Q10: **

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: **

Let be the transformation produced by a nonzero matrix with a zero determinant. What is the image of a unit square under ?

- Aanother square
- Ba line segment containing the origin
- Ca single point
- Da rhombus
- Ea parallelogram

**Q14: **

Let be a linear transformation of into itself with the property that and .

Using the fact that , find .

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Using the fact that , find .

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Find a vector so that .

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What is , where and ?

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By considering suitable linear combinations of and , find and .

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

Find the matrix which represents the linear transformation .

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**Q15: **

The vertex matrix of a square of side 1 shown is

Determine the vertex matrix of the image after a transformation by the matrix , and state what geometric figure it is.

- A, a parallelogram
- B, a rectangle
- C, a rhombus
- D, a square

**Q16: **

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

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