Linear Transformations,GeometryJay TreimanThe objective of this worksheet is to explore the geomerty of linear transformations. The first step is to write a linear transformation inits matrix representation. One can then explore the geomerty of linear transformations
<Text-field style="Heading 1" layout="Heading 1">Load the <Hyperlink linktarget="Help:LinearAlgebra" hyperlink="true"><Font bold="true" style="Hyperlink" size="18">LinearAlgebra</Font></Hyperlink> and <Hyperlink linktarget="Help:plots" hyperlink="true"><Font bold="true" style="Hyperlink" size="18">plots</Font></Hyperlink> packages.</Text-field>restart;
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<Text-field style="Heading 1" layout="Heading 1">The matrix representation </Text-field>of a linear transformation.The following is how one gets the matrix representation of the 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.One follows a general pattern. Here is anarbitrary transformation from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRkFGK0ZGRkE= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRkFGK0ZGRkE=. Thematrix is a 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn is the matrix. Multiplying bya two vector gives the coefficient positions.A := Matrix([[a,b],[c,d]]);
Multiply(A,Vector([x[1],x[2]]));This means that one has the coefficients LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=,and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMkYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=. Hence the transformation corresponds to the 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.Here is a check.X := Vector(2,symbol=x);
A := Matrix([[2,1],[1,2]]);
A.X;
<Text-field bookmark="areas" style="Heading 1" layout="Heading 1">The area of an image under a Linear Transformation.</Text-field>The general area change can be infered from what happensto a standard unit square with vertices [0,0], [0,1], [1,1],and [1,0]. Since a linear transformation takes parallel linesto parallel lines one can simply find the area of the newpolygon using the cross product, CrossProduct.Here LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn is applied to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNictSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMEYnRjQvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjRGNC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZURjQ= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNictSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjQvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjRGNC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZURjQ= to get the two new vectors that span the image of the unit square under LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn. The cross product of the vectors as vectors in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRkFGK0ZGRkE= gives the area.This can also be calculated using Determinant(A). (Confirmthis.)Here are the two vectors in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRkFGK0ZGRkE=, the norm of their corss product, and the determinant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn.v1 := convert(Multiply(A,Vector([1,0])),list):
v1 := Vector([op(v1),0]);
v2 := convert(Multiply(A,Vector([0,1])),list):
v2 := Vector([op(v2),0]);
Norm(CrossProduct(v1,v2),2);
Determinant(A);This fact about the area of transformed regions holds forany region in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRkFGK0ZGRkE= for which an area is defined. In otherwords, if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn is a region in the plane with area LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn isa linear transformation on the plane with matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn, then the area of 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 is 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.Here is an example of this. Consider the area bounded by the 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj4vJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRj5GK0ZYRj4=, transformed by the transformation 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.Here are plots of the two regions.A := Matrix([[1,1],[0,1]]);
plot([x^2,4],x=-2..2,color=[red,blue]);
c2 := convert(Multiply(A,Vector([t,t^2])),list);
c3 := convert(Multiply(A,Vector([t,4])),list);
plot([[op(c2),t=-2..2],[op(c3),t=-2..2]],color=[red,blue]);According to the formula, the area of the transformed area should be 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 timesare area of the original region. Here are the calculations from Calculus I, the area between curves, and the determinant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn.Area_1 := int(4-x^2,x=-2..2);
New_area := abs(Determinant(A))*Area_1;It should have the same area. To calculate this area one can take horizontal linesacross the region and integrate their lengths against LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn to get the area. (This isthe area between two functions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn.)In what follows the two functions x1(y) and 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 that define the newregion are found and the difference is integrated. Notethe use of allvalues to find the functions.solve({x=c2[1],y=c2[2]},{t,x});
solns_curves := allvalues(%);
x1 := subs(solns_curves[1],x);
x2 := subs(solns_curves[2],x);
New_area_1 := abs(int(x1-x2,y=0..4));The folrmula works in this case, as it will for all areas.
<Text-field style="Heading 1" layout="Heading 1">Plotting a polygon and animating </Text-field>a linear transformation.This is how one plots a polygon. This is the standard unit square. vertices1 :=[[0,0],[1,0],[1,1],[0,1]];
polygonplot(vertices1,view=[-2..2,-2..2],color=green);One can use the map function to use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn to transform the vertices.This enables one to plot the image of the unit square under LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn.Tsquare := map2(Multiply,A,map(Vector,vertices1));
Tsquare := map(convert,convert(Tsquare,list),list);
polygonplot(Tsquare,view=[-1..4,-1..4],color=blue);
<Text-field style="Heading 2" layout="Heading 2">Animation of a linear transformation.</Text-field>The following function produces an animation of a linear transformation.You do not need to know how it works. You should know that thereare other ways of doing this that may give different animations.
<Text-field style="Heading 3" layout="Heading 3">Animation procedure.</Text-field>LinTrans2d := proc(A::{matrix,listlist,Matrix})
local A1,T,t1,p1,listP,B,color1,list1,m;
if type(A,Matrix) then A1 := A;
else A1 := convert(A,Matrix); end if;
T := (A2,t) -> LinearAlgebra[Add](A2,IdentityMatrix(2),t,1-t);
list1 := map(Vector,[[0,0],[1,0],[1,1],[0,1]]);
p1 := plottools[polygon](map(convert,list1,list),color=blue,
linestyle=3,thickness=2):
listP := [p1]:
t1 :=0;
for t1 from 0 by 1/10 to 1 do
B := eval(T(A1,t1));
if LinearAlgebra[Determinant](B) >= 0 then color1 := `blue`;
else color1 := `red`; fi;
listP := [op(listP),plottools[polygon](
map(convert,map2(LinearAlgebra[Multiply],B,list1),list),color=color1)];
od:
print(plots[display](listP,
insequence=true,scaling=constrained));
end:Here is the procedure applied to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn. To see the animation, select the graph and then use the new tool bar to start the animation.LinTrans2d(A);
<Text-field style="Heading 1" layout="Heading 1">Summary</Text-field>
<Text-field style="_cstyle256" layout="Heading 1"><Font bold="true" size="18">Exercises</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">1</Text-field>Find the matrices corresponding to the following linear transformations
<Text-field style="Heading 3" layout="Heading 3">a</Text-field>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
<Text-field style="Heading 3" layout="Heading 3">b</Text-field>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
<Text-field style="Heading 3" layout="Heading 3">c</Text-field>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
<Text-field style="Heading 2" layout="Heading 2">2</Text-field>Find the area of the image of the unit square under LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn.
<Text-field style="Heading 2" layout="Heading 2">3</Text-field>Find the volume of the image of the unit cube under LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.
<Text-field style="Heading 2" layout="Heading 2">4</Text-field>Find the image of the vertices of standard unit square under LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn.
<Text-field style="Heading 2" layout="Heading 2">5</Text-field>Find the image of the unit cube uncer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.
<Text-field style="Heading 2" layout="Heading 2">6</Text-field>Plot the images of the standard unit square under LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYnand LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn. What do you think the transformations aredoing to the unit square.
<Text-field style="Heading 2" layout="Heading 2">7</Text-field>Animate what the linear transformations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvRjNRJ25vcm1hbEYn do tothe unit square.
<Text-field style="Heading 2" layout="Heading 2">8</Text-field>Do the animations change your conclusions about what thetransformations do to the unit square?
<Text-field style="Heading 2" layout="Heading 2">9</Text-field>
<Text-field style="Heading 2" layout="Heading 2">10</Text-field>
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