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In this lesson, we will learn how to identify reflection, rotation, and dilation matrices and how to use them to perform transformations of points on a plane.
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.
The point is translated using the vector . What are the coordinates of its image?
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 .
Using the fact that is perpendicular to , find the equation of .
Using the fact that , find the coordinates of and .
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 .
Q3:
A linear transformation of a plane sends the vector to . If the transformation is a rotation, where does it send ?
Q4:
Let 𝑇 ∈ 𝐿 ( 𝑉 ) be a linear transformation. Suppose the matrix for 𝑇 relative to a basis 𝐵 for 𝑉 is 𝑀 . Suppose 𝑃 is the transition matrix from another basis 𝐶 to 𝐵 . Determine the matrix for 𝑇 with respect to 𝐶 .
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.
Write a vector to represent the translation from Shape B to Shape C.
Write a vector to represent the translation from Shape C to Shape A.
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