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Lesson: Introduction to Linear Transformations

Worksheet • 5 Questions

Q1:

A translation of 3 units right and 2 units down can be described by the vector .

Describe the translation from point to point using a vector.

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  • E

The point is translated using the vector . What are the coordinates of its image?

  • A
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  • D

Q2:

Consider the given figure.

The points , , , and are corners of the unit square. This square is reflected in the line with equation to form the image .

As is the image of in the line through and , . Use this fact and the identity to find the gradient and hence equation of from the gradient of .

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Using the fact that is perpendicular to , find the equation of .

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Using the fact that , find the coordinates of and .

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

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line .

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Q3:

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

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Q4:

Let 𝑇 ∈ 𝐿 ( 𝑉 ) be a linear transformation. Suppose the matrix for 𝑇 relative to a basis 𝐡 for 𝑉 is 𝑀 . Suppose 𝑃 is the transition matrix from another basis 𝐢 to 𝐡 . Determine the matrix for 𝑇 with respect to 𝐢 .

  • A 𝑃 𝑀 𝑃 βˆ’ 1
  • B 𝑃 𝑀 𝑃 βˆ’ 1
  • C 𝑀 𝑃
  • D 𝑃 𝑀

Q5:

Shape A has been translated to Shape B and then to Shape C.

Write a vector to represent the translation from Shape A to Shape B.

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Write a vector to represent the translation from Shape B to Shape C.

  • A
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  • E

Write a vector to represent the translation from Shape C to Shape A.

  • A
  • B
  • C
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  • E
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