Section 2.4

Linear Transformations of the Plane ( T: R^2 -> R^2 )

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> read(`C:\\classes\\2002-2003\\spring\\math215\\Lectures\\LAprocs.map`);

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the name changecoords has been redefined

Warning, the name arrow has been redefined

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The Collapsing Transformations

- T: R^2 -> R^2 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]);

A := matrix([[0, 0], [0, 0]])

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Projection on x-axis.

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

A := matrix([[1, 0], [0, 0]])

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Projection on y-axis.

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

A := matrix([[0, 0], [0, 1]])

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Collapse to a single line y = 2x.

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

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

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What are the ranks of these matrices?

Are these matrices invertible?

****************************************************

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)]);

A := matrix([[cos(theta), -sin(theta)], [sin(theta)...

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Rotate the axis vector by a given angle

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

v := vector([1, 0])

> plotvectorrot(v,60);

[Maple Plot]

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

T( e[1] ) = ?

T( e[2] ) = ?

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]);

b1 := vector([1, 2])

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Pick a vector that is orthogonal to b1.

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

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 R^2 .

Write e[1] and e[2] in terms of the basis {b1,b2}.

Linear system 1:

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

vector([1, 0]) = c1*vector([1, 2])+c2*vector([-2, 1...

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Linear system 2:

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

vector([0, 1]) = c1*vector([1, 2])+c2*vector([-2, 1...

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Solve each linear system or solve the linear systems together:

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

Aug := matrix([[1, -2, 1, 0], [2, 1, 0, 1]])

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> Aug := rref(Aug);

Aug := matrix([[1, 0, 1/5, 2/5], [0, 1, -2/5, 1/5]]...

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

[1, 0] = 1/5*vector([1, 2])-2/5*vector([-2, 1])

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

[0, 1] = 2/5*vector([1, 2])+1/5*vector([-2, 1])

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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([1, 0]) = 1/5*T(vector([1, 2]))-2/5*T(vector([-2,...

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

T([0, 1]) = 2/5*T(vector([1, 2]))+1/5*T(vector([-2,...

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

T(vector([1, 2])) = vector([1, 2])

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

T(vector([-2, 1])) = -vector([-2, 1])

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

T([1, 0]) := vector([-3/5, 4/5])

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

T([0, 1]) := vector([4/5, 3/5])

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

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

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

[Maple Plot]

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Expansions, contractions, and shears

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

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

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

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

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

A := matrix([[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]);

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

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Moving the House

Create the house.

> plothouse();

[Maple Plot]

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Reflection about the line y = 2x.

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

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

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> transformhouse(A);

[Maple Plot]

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Rotation counterclockwise via 60^o .

> theta := 60*Pi/180;

theta := 1/3*Pi

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

A := matrix([[1/2, -1/2*sqrt(3)], [1/2*sqrt(3), 1/2...

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> transformhouse(A);

[Maple Plot]

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Expansions, contractions, and shears ?

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

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

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> transformhouse(A);

[Maple Plot]

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Exercises

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