Linear Transformations

<< LinearAlgebra`MatrixManipulation`

Part 1.  Matrix Transformations

A = {{1, 2, 1, -3, 0}, {2, 4, 1, -6, 1}, {-3, -6, 0, 
            9, -3}};
A // MatrixForm
u = {1, 0, 2, 1, 4}
v = {-3, 1, 2, 0, 1}

x = {1, 0, -2, 4}
y = {-3, 5, 1, 2}

Part 2.  Injectivity, Surjectivity and Isomorphisms

A = {{1, 2, 1, -3, 0}, {2, 4, 1, -6, 1}, {-3, -6, 0, 
            9, -3}};
A // MatrixForm
u1 = {-5, 2, 1, 0, -2}
u2 = {4, -1, 4, 2, 1}

Note that on the web page the following matrix is called C. (C is a Mathematica function, so we can't use it as a variable.)

C0 = {{2, -1}, {3, 2}, {1, 3}};
C0 // MatrixForm

Part 3.  The Geometry of 2 x 2 Matrix Maps

The following lines of Mathematica code define and plot a 10 by 10 grid of points in first quadrant of the plane.

Color[j_] := RGBColor[ .1*j, 1 -  .1*j,  .1*j]
Do[
        gridrow[j] = 
          ListPlot[Table[{i/10,  j/10},  {i,  0,  10}],  
            PlotStyle -> {Color[j],  PointSize[ .02]}, 
            PlotRange -> {{-3, 3}, {-3, 3}},  
            DisplayFunction -> Identity],  {j,  0,  10}];
Show[Table[gridrow[j],  {j,  0,  10}],  
        DisplayFunction -> $DisplayFunction,  
        AspectRatio -> Automatic];

The matrix T defines the linear transformation which will be applied to the grid points.

T = {{-1, 0}, {0, 1}};
T // MatrixForm

These lines apply the linear transformation to each grid point and plot the resulting points.

Do[
        transfrow[j] = 
          ListPlot[Table[(T . {i/10,  j/10}),  {i,  0,  10}],  
            PlotStyle -> {Color[j],  PointSize[ .02]}, 
            PlotRange -> {{-3, 3}, {-3, 3}},  
            DisplayFunction -> Identity],  {j,  0,  10}];
Show[Table[transfrow[j],  {j,  0,  10}],  
        DisplayFunction -> $DisplayFunction,  
        AspectRatio -> Automatic];

Part 4.  Matrix Transformations and Rigid Motions

Part 5.  Summary