**Section 2.3**

**Linear Transformations of Euclidean Spaces **

`> `
**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

`> `

**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?**

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

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:

=

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

**Examples of Linear Transformations**

**Example 1: **
Consider the mapping from
into

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

`> `

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]); **

`> `

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)]));**

`> `

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

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

`> `
**w:=evalm(k*v);**

`> `

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)]);**

`> `

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);**

`> `

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]); **

`> `

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);**

`> `

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)]);**

`> `

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]]);**

`> `

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

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

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

`> `

Choose any two vectors v and u in

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

`> `

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);**

`> `

Does T preserve addition?

B. Does T preserve scalar multiplication?

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

`> `

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);**

`> `

Does T preserve the operations of addition and scalar multiplication?

Take any two vectors in

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

`> `

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)]));**

`> `

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

**Matrix Representation of a Linear Transformation**

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

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( ).

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

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

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.

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

**Example 1 (Revisited): **
Consider the mapping from
into

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

`> `

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

`> `

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

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

`> `

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

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

`> `

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

`> `

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

`> `

**Properties of Linear Transformations**

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

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.

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

**Visualizing Linear Transformations**

Car parked in the garage.

`> `

`> `
**plotcar();**

`> `

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);**

`> `

** **

**Exercises**

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