**Section 2.4**

**Linear Transformations of the Plane ( T: **
** ->**
** ) **

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

Projections to a single point 0.

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

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Projection on x-axis.

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

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Projection on y-axis.

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

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Collapse to a single line y = 2x.

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**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 **

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

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Rotate the axis vector by a given angle

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

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**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.

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**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:

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

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Linear system 2:

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

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Solve each linear system or solve the linear systems together:

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

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**Aug := rref(Aug); **

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

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**[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,

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

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

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**T(evalm(b1)) = evalm(b1);**

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

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

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**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 )__

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

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__Vertical Expansion (r >1) Contraction (r < 0 )__

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

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__Horizontal Shear__

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

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__Vertical Shear__

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

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**Moving the House**

Create the house.

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**plothouse();**

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Reflection about the line y = 2x.

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**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 .

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**theta := 60*Pi/180;**

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**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.