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Homework Help

Lesson: Matrix of a Linear Transformation

In this lesson, we will learn how to find the matrix of a linear transformation and how to describe it geometrically.

Worksheet • 14 Questions

Q1:

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

Q2:

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

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 βˆ’ 2 𝑐 βˆ’ 2 𝑑 𝑒 𝑓 1 βˆ’ 2 𝑒 βˆ’ 2 𝑓 
  • C  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 
  • D  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 
  • E  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 

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

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

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

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

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

What can you say about the vector 𝐴 𝑀 ?

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

Q3:

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