Worksheet: Matrix of a Linear Transformation

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.

• A
• B
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• E

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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• E

Q7:

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

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• E

Q8:

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

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• E

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.

• A
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Q10:

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

• A
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• E

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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