# Worksheet: Introduction to Linear Transformations

In this worksheet, we will practice identifying reflection, rotation, and dilation matrices and using them to perform transformations of points on a plane.

Q1:

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

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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 .

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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:

Let be a linear transformation. Suppose the matrix for relative to a basis for is . Suppose is the transition matrix from another basis to . Determine the matrix for with respect to .

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Q4:

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

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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.

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

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