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). You ll recognizethis right away as a reflection across is special to this operator as it s the set of fixed points. In other words,it s the 1-eigenspace. They-axis is also special as every point (0,y) is sent to its negation (0,y). That means they-axis is an eigenspace with eigenvalue 1, that is, it s the 1-eigenspace.
Logarithmic spirals whose equations in polar coordinates r= c2 2 =ˇare invariant subsets of this rotary contraction, where cis any constant. Transformations of R3. A 3 3 matrix describes a transformation of space, that is, a 3-D operator. There are many kinds of such transformations, some isometries, some not. Isometries include (1) re
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