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));

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

S:=rand(-10..10)(): 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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}: imagepoints:=pointplot(setofimages,color=black): imageline:=plot([t,S*t,t=-120..120],color=red): NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNictSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRK01vbm9zcGFjZWRGKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJXRydWVGKC8lJ2l0YWxpY0dGOC8lKnVuZGVybGluZUdRJmZhbHNlRigvJSpzdWJzY3JpcHRHRj0vJSxzdXBlcnNjcmlwdEdGPS8lK2ZvcmVncm91bmRHUSpbMjU1LDAsMF1GKC8lK2JhY2tncm91bmRHUShbMCwwLDBdRigvJSdvcGFxdWVHRj0vJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRj0vJSljb21wb3NlZEdGPS8lKmNvbnZlcnRlZEdGPS8lK2ltc2VsZWN0ZWRHRj0vJSxwbGFjZWhvbGRlckdGPS8lMGZvbnRfc3R5bGVfbmFtZUdRLE1hcGxlfklucHV0RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRy8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYjLUYtNjlRKmRpc3BsYXkzZEYoRjBGM0Y2L0Y6Rj1GO0Y+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduL0ZqblElYm9sZEYoRlxvLUYtNjlGLy9GMVEwVGltZXN+TmV3flJvbWFuRihGMy9GN0Y9RjlGO0Y+RkAvRkNGRy9GRlEuWzI1NSwyNTUsMjU1XUYoRkhGSkZMRk5GUEZSRlQvRldRKTJEfklucHV0RigvRlpGRy9GZm5GXXAvRmhuRmlvL0ZqblEnaXRhbGljRihGXG8tRiQ2JUZmby1JKG1mZW5jZWRHRiU2Iy1GJDYoLUYtNjlGL0YwRjNGNkZjb0Y7Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GZG9GXG8tRmhwNiUtRiQ2JS1GLTY5USxpbWFnZXBvaW50c0YoRjBGM0Y2RmNvRjtGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZkb0Zcby1JI21vR0YlNjNRIixGKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRj0vJSpzZXBhcmF0b3JHRjgvJSdsc3BhY2VHUSQwZW1GKC8lJ3JzcGFjZUdRM3Zlcnl0aGlja21hdGhzcGFjZUYoLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lMGZvbnRfc3R5bGVfbmFtZUdGWC8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRy1GLTY5USppbWFnZWxpbmVGKEYwRjNGNkZjb0Y7Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GZG9GXG8vJSVvcGVuR1EifGZyRigvJSZjbG9zZUdRInxockYoLUZmcTYzRmhxRmlxRlxyRl5yRmByRmNyRmZyRmhyRmpyRl1zRmBzRmJzRmRzL0Znc0ZfcEZocy9GW3RGRy9GXXRGXXAtRi02OVEoc2NhbGluZ0YoRmhvRjNGam9GOUY7Rj5GQEZbcEZccEZIRkpGTEZORlBGUkZURl5wRmBwRmFwRmJwRmNwRlxvLUZmcTYzUSI9RihGaXFGXHIvRl9yRj0vRmFyUS90aGlja21hdGhzcGFjZUYoL0ZkckZkdUZmckZockZqckZdc0Zgc0Zic0Zkc0ZpdEZoc0ZqdEZbdS1GLTY5USxjb25zdHJhaW5lZEYoRmhvRjNGam9GOUY7Rj5GQEZbcEZccEZIRkpGTEZORlBGUkZURl5wRmBwRmFwRmJwRmNwRlxvLUZmcTYzUSI7RihGaXFGXHJGXnJGYHJGZXVGZnJGaHJGanJGXXNGYHNGYnNGZHNGaXRGaHNGanRGW3VGLDcjQyQtSSpkaXNwbGF5M2RHRig2JDwkSSxpbWFnZXBvaW50c0dGKEkqaW1hZ2VsaW5lR0YoL0koc2NhbGluZ0dGKEksY29uc3RyYWluZWRHRigiIiI=

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

Questions: Why do the points only cover part of the line? What would you need to change to get points on the uncovered portion 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). How many of these points are mapped to zero, i.e. in the null space? In defining the set, "setofimages", the points in the null space were excluded. So lets count the number of points left in "setofimages". 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

Notice that none of our image points were in the null space. To be in the null space a point, X, must satisfy the following matrix equation AX = 0. Or in our case eq1:=Row(A,1).Vector(<x,y,z>)=0;

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Why were none of our points in the null space? Remember that x, y and z were all defined as integer values between -10 and 10. Lets plot some points in the nullspace by computing some random values of y and z and use these to find the appropriate value of x. 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 nullpoints:=NULL: for i from 1 to 300 do: valueofy := rand(-10..10)(): valueofz := rand(-10..10)(): nullpoints := nullpoints union {Vector(3,[eval(eq2,[y=valueofy,z=valueofz]),valueofy,valueofz])}: end do: plotofpoints:=pointplot3d(nullpoints,view=[-15..15,-15..15,-15..15],symbol=circle,color=black): plotofplane:=implicitplot3d(eq1,x=-15..15,y=-15..15,z=-15..15,color=red,style=patchnogrid,axes=normal,transparency=.5): display3d(plotofpoints,plotofplane);

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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.