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
  • C
  • D
  • E

Where does this transformation map and ?

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q2:

Suppose the linear transformation 𝐿 sends ( 1 , 0 ) to ( 1 , 5 ) and ( 1 , 1 ) to ( 6 , 6 ) . What is the absolute value of the determinant of the matrix representing 𝐿 ?

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 1 .

Q4:

The determinant of a 2 × 2 matrix is 1 . 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 𝐴 , 𝐵 , 𝐶 , and 𝐷 onto 𝐴 , 𝐵 , 𝐶 , and 𝐷 , as shown.

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

  • A 6 5
  • B 8 5
  • C 6 5
  • D 8 5
  • E 9 5

Q7:

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

  • A
  • B
  • C
  • D
  • E

Q8:

What is the matrix 𝑀 that sends points 𝐴 , 𝐵 , and 𝐶 to 𝐴 , 𝐵 , and 𝐶 as shown?

  • A 𝑀 = 1 2 3 4
  • B 𝑀 = 1 1 3 1
  • C 𝑀 = 3 1 2 4
  • D 𝑀 = 1 3 4 1
  • E 𝑀 = 2 3 4 1

Q9:

A linear transformation is formed by rotating every vector in 2 through an angle of 𝜋 4 and then reflecting the resulting vector in the 𝑥 -axis. Find the matrix of this linear transformation.

  • A 2 2 2 2 2 2 2 2
  • B 2 2 2 2 2 2 2 2
  • C 2 2 0 0 2 2
  • D 2 2 2 2 2 2 2 2
  • E 2 2 2 2 2 2 2 2

Q10:

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

  • A
  • B
  • C
  • D
  • 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 ?

  • A
  • B
  • C
  • D
  • E

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

  • A
  • B
  • C
  • D
  • E

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 ?

  • A
  • B
  • C
  • D
  • E

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

  • A
  • B
  • C
  • D
  • E

What can you say about the vector ?

  • A
  • B
  • C
  • D
  • E

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

  • A
  • B
  • C
  • D
  • E

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