Section 2.3

Linear Transformations of Euclidean Spaces

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Review of functions

A function f: X -> Y is a rule that associates with each x in the set X an element y = f(x) in Y. We

say f maps the set X into the set Y.

- X is the domain of f(x)

- Y is the codomain of f(x)

- f(X) is the range of f(x)

If H is a subset of X then the image of H under f is f(H).

If K is a subset of Y then the inverse image of K under f is (K)

What is a Linear Transformation?

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A linear transformation , T, is a mapping from a vector space V into a vector space W
satisfying two conditions:

. ( T preserves addition): for any and in V

=

. ( T preserves scalar multiplication): for any v in V and any scalar k:

=

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Examples of Linear Transformations

Example 1: Consider the mapping from into

> T :=(x,y)->(y,x);

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Does T preserve addition? That is, is T(v + u) = T(v) + T(u) for any two vectors v and u in ?

Take any two vectors in :

> v:=vector([a,b]); u:=vector([c,d]);

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Compute the action of T on v + u

> `T(v + u)`=[T(a+c,b+d)];

and the sum T(v) + T(u)

> `T(v)`=[T(a,b)];`T(u)`:=[T(c,d)];

> `T(v)+T(u)`= evalm(vector([T(a,b)])+ vector([T(c,d)]));

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Is T(v + u) = T(v) + T(u)?

Does T preserve scalar multiplication? If k is any real scalar

> w:=evalm(k*v);

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compare the action of T on k*v, that is T( ) to :

> `T(kv)`=[T(k*a,k*b)];
`kT(v)`=evalm(k*[T(a,b)]);

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Is T( ) = ?

This example shows that the transformation T preserves addition and scalar multiplication.
T is a "
linear transformation ".

Example 2: Consider the mapping T from to given by

> T :=(x,y,z)->(x-y,y-z);

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Does T preserve addition? That is, is T(v + u) = T(v) + T(u) for any two vectors u and v in ?

Take any two vectors in

> v:=vector([a1,a2,a3]); u:=vector([b1,b2,b3]);

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Compare the action of T on v + u to the sum of T(v) and T(u):

> `T(v+u)`=[T(a1+b1,a2+b2,a3+b3)];

> `T(v)`=[T(a1,a2,a3)];
`T(u)`=[T(b1,b2,b3)];

Compute

> `T(v)+T(u)`= evalm(vector([T(a1,a2,a3)])+ vector([T(b1,b2,b3)]));

Is T(v + u) = T(v) + T(u)?

Does the transformation T preserve scalar multiplication? For any scalar k

> w:=evalm(k*v);

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Compare the action of T on , that is T( ) to :

> `T(kv)`=[T(k*a1,k*a2,k*a3)];
`kT(v)`=evalm(k*[T(a1,a2,a3)]);

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Is T( ) = ?

Example 3: Does matrix multiplication define a linear transformation?

Choose a matrix and an appropriate vector

> A:=matrix([[1,3],[-2,3]]);
X:=matrix([[x],[y]]);

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Define T : ---> in terms of the matrix multiplication of A and X

> T:=X->multiply(A,X);

> `T(X)` =T(X);

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Choose any two vectors v and u in

> v:=matrix([[a],[b]]);u:=matrix([[c],[d]]);

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A. Determine the action of T on v + u and compare it to T(v) + T(u)

> `T(v) + T(u)`=evalm(T(v)+T(u));
`T(v + u )`=T(v+u);

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B. Does T preserve scalar multiplication?

> `kT(v)`=evalm(k*T(v));
`T(kv)`=T(evalm(k*v));

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Therefore, matrix multiplication is an example of a linear transformation.

Let us consider another type of transformations

Example 4: Consider a transformation T on

> T :=(x,y)->(x-y,y^2);

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Does T preserve the operations of addition and scalar multiplication?

Take any two vectors in

> v:=vector([a,b]); u:=vector([c,d]);

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Is T(v + u) = T(v) + T(u)?

> `T(v+u)`=[T(a+c,b+d)];

> `T(v)+T(u)`= evalm(vector([T(a,b)])+vector([T(c,d)]));

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Compare T(v + u) and T(v) + T(u). Are they equal?

Matrix Representation of a Linear Transformation

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Let T : -> be a linear tranformation, and let B = { , , . . ., } be a

basis for . For any vector v in , the vector T(v) is uniquely determined

by the vectors T( ), T( ), . . ., T( ).

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Let T : -> be a linear tranformation, and let A be an m x n matrix whose

jth column vector is T( ), which we denote as,

A = [ T( ) T( ) . . . T( ) ]

Then T(x) = Ax for each column vector x in . A is the standard matrix representation.

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Example 1 (Revisited): Consider the mapping from into

> T :=(x,y)->(y,x);

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> a1 := T(1,0); a2:=T(0,1);

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> A := transpose(matrix(2,2,[a1,a2]));

> evalm(A)*matrix(2,1,[x,y]) = evalm(A&*matrix(2,1,[x,y]));

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Example 2 (Revisited): Consider the mapping T from to given by

> T :=(x,y,z)->(x-y,y-z);

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> a1 := T(1,0,0); a2:=T(0,1,0); a3:=T(0,0,1);

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> A := transpose(matrix(3,2,[a1,a2,a3]));

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Properties of Linear Transformations

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Let T: -> be a linear transformation with a standard matrix representation A.

The kernal of T, denoted ker(T) is the set of all solutions to T(x) = 0. Note this is the

same as computing all solutions to Ax = 0. Hence, the kernal of T is the null space of A.

The range of T is the column space of A.

The dim(range(T)) called the rank of T is the dimension of the column space of A.

The dim(ker(T)) called the nullity of T is the dimension of the null space of A.

A linear transformation is invertible if and only if its associated matrix is invertible.

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Visualizing Linear Transformations

Car parked in the garage.

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> plotcar();

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We can use a linear transformations to rotate our car 90 degrees.

We start with a matrix that rotates objects by 90 degrees in the y-z plane (that is, around the x-axis). Such a transformation should send e1 to e1, e2 to e3, and e3 to -e2. We designate the matrix rot1.

> rot1:= matrix(3,3,[[1,0,0],[0,0,-1],[0,1,0]]);

> plotcar2(rot1);

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Exercises

1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 21, 23, 25.