Linear Transformations and matrices
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Enlargements, shears and rotations
There are some special cases of [Equation goes here - download the original to see it.] matrices. Enlargement by a factor k This has matrix representation [Equation goes here - download the original to see it.] Example Sketch the image of the square with vertices at the origin, and at [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] under the matrix [Equation goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] , [Equation goes here - download the original to see it.] , [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Stretches: A stretch parallel to the x-axis is represented by the matrix. [Equation goes here - download the original to see it.] A stretch parallel to the y-axis is represented by the matrix [Equation goes here - download the original to see it.] Shears. A shear parallel to the x-axis is represented by the matrix [Equation goes here - download the original to see it.] A shear parallel to the y-axis is represented by the matrix [Equation goes here - download the original to see it.] Rotations A rotation about the origin through the angle q anticlockwise has a special matrix form. This is [Equation goes here - download the original to see it.] To show this we employ trigonometry. Imagine a square with vertices at the origin, and at [Equation goes here - download the original to see it.] rotated about the origin through an angle q. As the diagram shows, the image of under this transformation would be [Equation goes here - download the original to see it.] , and the image of [Equation goes here - download the original to see it.] under this transformation would be [Equation goes here - download the original to see it.]. Since in the matrix [Diagram goes here - download the original to see it.] In the matrix [Equation goes here - download the original to see it.] the vectors [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it. represent the images of [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] under M respectively, this gives the rotation matrix as [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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