Linear Transformations and matrices
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Reflections
The reflection in the line with angle q has matrix representation [Equation goes here - download the original to see it.] To show this we must again apply geometric intuition and use some trigonometry. [Diagram goes here - download the original to see it.] In this diagram the line represents the line of reflection with angle q to the x-axis. The arrows represent the process of reflecting the point [Equation goes here - download the original to see it.] about this line. OAB is a right-angled triangle, with hypotenuse of length 1 and angle q, so the length AB is . BC is twice this length, so it is [Equation goes here - download the original to see it.]. We need to find the lengths x and y. In the triangle BCD we have [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] This being an application of the trigonometric identity, [Equation goes here - download the original to see it.]. Also in the triangle BCD we have [Equation goes here - download the original to see it.] This also requires knowledge of trigonometric identities. This value of y gives the size of the y-coordinate of the point C but this point lies below the x axis, and hence the coordinates of C are [Equation goes here - download the original to see it.] That is the image of [Equation goes here - download the original to see it.] under the reflection through the line with angle q is[Equation goes here - download the original to see it.]. By a similar argument we can show that the image of under this reflection is[Equation goes here - download the original to see it.], so the matrix representing this transformation is [Equation goes here - download the original to see it.]
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Contents of
Linear Transformations and matrices
1 Matrix Transformations
2 Enlargements, shears and rotations
3 Reflections
4 Combinations of rotations and reflections
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