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Worksheet: Matrix of a Linear Transformation

Q1:

Consider the linear transformation which maps ( 1 , 1 ) to ( 3 , 7 ) and ( 2 , 0 ) to ( 2 , 6 ) .

Find the matrix 𝐴 which represents this transformation.

  • A 𝐴 =  3 1 4 3 
  • B 𝐴 =  1 2 3 4 
  • C 𝐴 =  2 3 1 4 
  • D 𝐴 =  1 2 3 4 
  • E 𝐴 =  3 2 4 1 

Where does this transformation map ( 1 , 0 ) and ( 0 , 1 ) ?

  • A ( 1 , 0 ) β†’ ( 1 , 3 ) , ( 0 , 1 ) β†’ ( 2 , 4 )
  • B ( 1 , 0 ) β†’ ( 2 , 3 ) , ( 0 , 1 ) β†’ ( 1 , 4 )
  • C ( 1 , 0 ) β†’ ( 3 , 1 ) , ( 0 , 1 ) β†’ ( 4 , 3 )
  • D ( 1 , 0 ) β†’ ( 1 , 3 ) , ( 0 , 1 ) β†’ ( 3 , 4 )
  • E ( 1 , 0 ) β†’ ( 3 , 2 ) , ( 0 , 1 ) β†’ ( 4 , 1 )

Q2:

The determinant of a 2 Γ— 2 matrix is βˆ’ 1 . What is the area of the image of a unit square under the transformation it represents?

Q3:

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

Q4:

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
  • B
  • C
  • D
  • E

Q5:

A linear transformation maps the points 𝐴 , 𝐡 , 𝐢 , and 𝐷 onto 𝐴 βˆ— , 𝐡 βˆ— , 𝐢 βˆ— , 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 βˆ’ 6 5
  • D βˆ’ 8 5
  • E 9 5

Q6:

The matrix 𝐴 represents a linear transformation which sends the vector  1 0  to  𝑝 π‘ž  . What can you say about the matrix 𝐴 ?

  • AIts first row is [ 𝑝 π‘ž ] .
  • BIts second column is  𝑝 π‘ž  .
  • CIts second row is [ 𝑝 π‘ž ] .
  • DIts first column is  𝑝 π‘ž  .

Q7:

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 𝐴 which represents a rotation of ℝ 3 by 9 0 ∘ about an axis through the origin and in the direction of

What is 𝐴 𝑛 ?

  • A  1 0 0 
  • B βˆ’ 𝑛
  • C  0 1 0 
  • D 𝑛
  • E  0 0 1 

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

  • A  π‘Ž 𝑏 2 βˆ’ 2 π‘Ž βˆ’ 2 𝑏 𝑐 𝑑 2 βˆ’ 2 𝑐 βˆ’ 2 𝑑 𝑒 𝑓 1 βˆ’ 2 𝑒 βˆ’ 2 𝑓 
  • B  π‘Ž 𝑏 βˆ’ 2 π‘Ž βˆ’ 2 𝑏 𝑐 𝑑 1 βˆ’ 2 𝑐 βˆ’ 2 𝑑 𝑒 𝑓 βˆ’ 2 𝑒 βˆ’ 2 𝑓 
  • C  π‘Ž 𝑏 1 βˆ’ 2 π‘Ž βˆ’ 2 𝑏 𝑐 𝑑 βˆ’ 2 𝑐 βˆ’ 2 𝑑 𝑒 𝑓 βˆ’ 2 𝑒 βˆ’ 2 𝑓 
  • D  π‘Ž 𝑏 βˆ’ 2 βˆ’ 2 π‘Ž βˆ’ 2 𝑏 𝑐 𝑑 βˆ’ 2 βˆ’ 2 𝑐 βˆ’ 2 𝑑 𝑒 𝑓 βˆ’ 1 βˆ’ 2 𝑒 βˆ’ 2 𝑓 
  • E  π‘Ž 𝑏 βˆ’ 2 π‘Ž βˆ’ 2 𝑏 𝑐 𝑑 βˆ’ 2 𝑐 βˆ’ 2 𝑑 𝑒 𝑓 1 βˆ’ 2 𝑒 βˆ’ 2 𝑓 

The vector is perpendicular to 𝑛 . What can you say about the direction of the vector 𝑀 = 𝐴 𝑣 ?

  • A 𝑀 will be perpendicular to 𝑣 but not necessarily to 𝑛
  • B 𝑀 will be parallel to 𝑛
  • C 𝑀 will be perpendicular to both 𝑛 and 𝑣
  • D 𝑀 will be parallel to 𝑣
  • E 𝑀 will be perpendicular to 𝑛 but not necessarily to 𝑣

What can you say about the magnitude of 𝑀 = 𝐴 𝑣 ?

  • A | 𝑀 | = | 𝑣 |
  • B | 𝑀 | = 3 | 𝑛 |
  • C | 𝑀 | = 3 | 𝑣 |
  • D | 𝑀 | = | 𝑛 |
  • E | 𝑀 | = 1

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

  • A  1 1 4 
  • B  3 3 βˆ’ 1 2 
  • C  βˆ’ 3 3 0 
  • D  1 1 βˆ’ 4 
  • E  3 βˆ’ 3 1 2 

What can you say about the vector 𝐴 𝑀 ?

  • A 𝐴 𝑀 = 𝑛
  • B 𝐴 𝑀 = 𝑀
  • C 𝐴 𝑀 = 𝑣
  • D 𝐴 𝑀 = βˆ’ 𝑣
  • E 𝐴 𝑀 = βˆ’ 𝑛

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

  • A 1 5 4  2 3 1 2 βˆ’ 3 1 1 0 βˆ’ 4 
  • B 1 9  4 1 8 7 4 βˆ’ 4 βˆ’ 4 8 1 
  • C  4 1 8 7 4 βˆ’ 4 βˆ’ 4 8 1 
  • D  2 3 1 2 βˆ’ 3 1 1 0 βˆ’ 4 
  • E  2 2 1 3 βˆ’ 3 0 1 1 βˆ’ 4 

Q8:

Consider the linear transformations for which v , the image of  1 0  , and w , the image of  0 1  , are unit vectors. Let 𝐿 be a linear transformation of this kind which has the additional property that the area of the parallelogram with vertices 0 , v , w , and v w + is as big as possible. What are the possible values of the measure of the angle between v and w for the transformation 𝐿 ?

  • A 2 7 0 ∘ only
  • B 9 0 ∘ only
  • C 0 ∘
  • D 9 0 ∘ and 2 7 0 ∘
  • E 1 8 0 ∘

Q9:

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

  • A 𝑏 = 0 , β„Ž = 0 , 𝑑 = 0 , 𝑓 = 0 , 𝑒 = 1
  • B 𝑐 = 1 , 𝑓 = 1 , 𝑔 = 1 , β„Ž = 1 , 𝑖 = 0
  • C 𝑏 = 1 , β„Ž = 1 , 𝑑 = 1 , 𝑓 = 1 , 𝑒 = 0
  • D 𝑐 = 0 , 𝑓 = 0 , 𝑔 = 0 , β„Ž = 0 , 𝑖 = 1
  • E 𝑐 = 1 , 𝑓 = 1 , 𝑔 = 0 , β„Ž = 0 , 𝑖 = 1

Q10:

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

  • A 𝑐 = 0 , 𝑓 = 0 , 𝑔 = 0 , β„Ž = 0 , 𝑖 = 1
  • B 𝑏 = 1 , β„Ž = 1 , 𝑑 = 1 , 𝑓 = 1 , 𝑒 = 0
  • C 𝑏 = 0 , β„Ž = 1 , 𝑑 = 0 , 𝑓 = 0 , 𝑒 = 0
  • D 𝑏 = 0 , β„Ž = 0 , 𝑑 = 0 , 𝑓 = 0 , 𝑒 = 1
  • E 𝑏 = 0 , β„Ž = 0 , 𝑑 = 0 , 𝑓 = 0 , 𝑒 = 0

Q11:

Find the matrix of the transformation that maps the points 𝐴 , 𝐡 , 𝐢 , and 𝐷 onto 𝐴 βˆ— , 𝐡 βˆ— , 𝐢 βˆ— , and 𝐷 βˆ— as shown.

  • A ⎑ ⎒ ⎒ ⎣ βˆ’ 9 5 1 1 1 5 3 5 βˆ’ 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 9 5 1 1 1 5 3 5 βˆ’ 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ βˆ’ 5 9 1 5 1 1 3 5 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎣ βˆ’ 9 5 1 1 1 5 βˆ’ 3 5 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ βˆ’ 9 5 βˆ’ 3 5 1 1 5 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦

Q12:

What is the matrix 𝑀 that sends points 𝐴 , 𝐡 , and 𝐢 to 𝐴 βˆ— , 𝐡 βˆ— , and 𝐢 βˆ— 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 

Q13:

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 βˆ’ 1 .

Q14:

A linear transformation of a plane sends vector  1 0  to  𝑝 π‘ž  . If the transformation is a rotation, where does it send  0 1  ?

  • A  π‘ž 𝑝 
  • B  π‘ž βˆ’ 𝑝 
  • C  𝑝 π‘ž 
  • D  βˆ’ π‘ž 𝑝 
  • E  βˆ’ π‘ž βˆ’ 𝑝 