Transcription of Some linear transformations on R2 Math 130 Linear Algebra
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Some linear transformations on R2 Math 130 Linear Algebra
Some Linear transformations onR2 Math 130 Linear AlgebraD Joyce, Fall 2015 Let s look at some some Linear transformations on the planeR2. We ll look at severalkinds of operators onR2including reflections, rotations, scalings, and ll illustrate these transformations by applying them to the leaf shown in figure 1. Ineach of the figures thex-axis is the red line and they-axis is the blue 1: Basic leafFigure 2: Reflected acrossx-axisExample 1(A reflection).Consider the 2 2 matrixA=[100 1]. Take a generic pointx= (x,y) in the plane, and write it as the column vectorx=[xy]. Then the matrix productAxisAx=[100 1][xy]=[x y]Thus, the matrixAtransforms the point (x,y) to the pointT(x,y) = (x, y).
Example 8 (Projections). Consider the 2 2 matrix 1 0 0 0 . It sends the vector x y to the vector x 0 . This particular transformation is a projection from 2-space onto the x-axis that forgets the y-coordinate. Projections have 1-eigenspaces and 0-eigenspaces. What are they for this projection? This projection will map our leaf to the x-axis as ...
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