Image of a Linear Transformation Worksheet by Russell Blyth restart: with(LinearAlgebra): with(plots): with(plottools):

Warning, the name changecoords has been redefined

Warning, the assigned name arrow now has a global binding

The basic objectives are: 1) Investigate the image of a particular linear transformation. 2) Graphically investigate the null space of the linear transformation.

Define a random 2 x 3 matrix of rank 1. A := RandomMatrix(1,3,generator=rand(-10..10)); A := <A,rand(-10..10)()*Row(A,1)>;

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Interpret A as a matrix that represents a linear transformation T from 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 to 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 What is the dimension of the image of T? We investigate by creating 100 random points in 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 and finding and plotting the images of these 100 points under multiplication on the left by the matrix A. First the 100 points: setofpoints := {seq(Vector(3,[rand(-10..10)(),rand(-10..10)(),rand(-10..10)()]), i=1..100)}: pointplot3d(setofpoints,view=[-10..10,-10..10,-10..10], axes=normal,color=black);

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Rotate the plot to see that the points are spread around in 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. Next, plot the images of these points: setofimages := {seq(A.setofpoints[i],i=1..100)} minus {0}: pointplot(setofimages,color=black);

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Questions: why do you see only a line segment for the image rather than a whole line? How can you get more of the line? What is the slope of the line (note that the line does not have the slope it appears to have due to the scaling of the axes). Next, look at the null space of A. nullbasis:=NullSpace(A);

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

Plot a few points in the nullspace by computing some random linear combinations of the basis elements. nullpoints := {seq(rand(-10..10)()*nullbasis[1] + rand(-10..10)()*nullbasis[2],i=1..500)} minus {0}: pointplot3d(nullpoints,view=[-10..10,-10..10,-10..10], axes=normal,color=black);

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

Rotate the plot to see that the null space is a plane, and hence has dimension 2. Exercise: 1) Create a random 3x3 matrix B of rank 2. Repeat the calculations and plots performed above for the matrix A for your matrix B. Note that B represents a linear transformation from 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, so all plots will be 3D plots.