Section 2.4

Linear Transformations of the Plane ( T: -> )

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The Collapsing Transformations

- T: -> can be represented by T(x) = Ax, where A is some 2 x 2 matrix.

Projections to a single point 0.

> A := matrix(2,2,[0,0,0,0]);

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Projection on x-axis.

> A:=matrix(2,2,[1,0,0,0]);

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Projection on y-axis.

> A:=matrix(2,2,[0,0,0,1]);

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Collapse to a single line y = 2x.

> A:=matrix(2,2,[1,-3,2,-6]);

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What are the ranks of these matrices?

Are these matrices invertible?

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Projection onto a line must map every vector onto the line and

every vector along the line is left fixed.

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Invertible Linear Transformations of the Plane

Rotation counterclockwise through q

> A:=matrix(2,2,[cos(theta),-sin(theta),sin(theta),cos(theta)]);

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Rotate the axis vector by a given angle

> v := vector([1,0]);

> plotvectorrot(v,60);

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Let T be a linear transformation that rotates vector counterclockwise through q.

T( ) = ?

T( ) = ?

Reflection about a line through the origin

Let y = 2x.

Pick a vector that lies on the line.

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> b1 := vector([1,2]);

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Pick a vector that is orthogonal to b1.

> b2 := vector([-2,1]);

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Reflect about the line y=2x , then

1.) T(b1) = b1.

2.) T(b2) = -b2.

Note that {b1,b2} forms a basis for .

Write and in terms of the basis {b1,b2}.

Linear system 1:

> vector([1,0]) = c1*evalm(b1) + c2*evalm(b2);

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Linear system 2:

> vector([0,1]) = c1*evalm(b1) + c2*evalm(b2);

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Solve each linear system or solve the linear systems together:

> Aug := augment(b1,b2,[1,0],[0,1]);

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> Aug := rref(Aug);

> [1,0] = Aug[1,3]*evalm(b1) + Aug[2,3]*evalm(b2);

> [0,1] = Aug[1,4]*evalm(b1) + Aug[2,4]*evalm(b2);

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Apply the linear transformation T to both sides of the equations to get,

> T([1,0]) = Aug[1,3]*T(evalm(b1)) + Aug[2,3]*T(evalm(b2));

> T([0,1]) = Aug[1,4]*T(evalm(b1)) + Aug[2,4]*T(evalm(b2));

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> T(evalm(b1)) = evalm(b1);

> T(evalm(b2)) = - evalm(b2);

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> T([1,0]) := evalm(Aug[1,3]*b1 - Aug[2,3]*b2);

> T([0,1]) := evalm(Aug[1,4]*b1 - Aug[2,4]*b2);

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> A := augment(T([1,0]),T([0,1]));

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> plotvectorref([1,1],A,2*x);

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Expansions, contractions, and shears

Horizontal Expansion (r >1) Contraction (r < 0 )

> A := matrix(2,2,[r,0,0,1]);

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Vertical Expansion (r >1) Contraction (r < 0 )

> A := matrix(2,2,[1,0,0,r]);

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Horizontal Shear

> A := matrix(2,2,[r,1,0,1]);

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Vertical Shear

> A := matrix(2,2,[1,0,1,r]);

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Moving the House

Create the house.

> plothouse();

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Reflection about the line y = 2x.

> A:= matrix(2,2,[-3/5,4/5,4/5,3/5]);

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> transformhouse(A);

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Rotation counterclockwise via .

> theta := 60*Pi/180;

> A:=matrix(2,2,[cos(theta),-sin(theta),sin(theta),cos(theta)]);

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> transformhouse(A);

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Expansions, contractions, and shears ?

> A := matrix(2,2,[5,0,0,1]);

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> transformhouse(A);

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Exercises

1, 2, 3, 6, 7, 8, 10-15.