c2s4 - 2.4 Linear Transformations of the Plane 1 Chapter 2...

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2.4 Linear Transformations of the Plane 1 Chapter 2. Dimension, Rank, and Linear Transformations 2.4 Linear Transformations of the Plane (in brief) Note. If A is a 2 × 2 matrix with rank 0 then it is the matrix A = 0 0 0 0 and all vectors in R 2 are mapped to 0 under the transformation with asociated matrix A (We can view 0 as a 0 dimensional space). If the rank( A ) = 1, then the column space of A , which is the range of T A , is a one dimensional subspace of R 2 . In this case, T A projects a vector onto the column space. See page 155 for details. Note. We can rotate a vector in R 2 about the origin through an angle θ by applying T A where A = cos θ - sin θ sin θ cos θ . This is an example of a rigid transformation of the plane since lengths are not changed under this transformation.
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2.4 Linear Transformations of the Plane 2 Note. We can reflect a vector in R 2 about the x -axis by applying T X where X = 1 0 0 - 1 .
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