Transcript A x

m
4.2 Linear Transformations from Rn to R
Functions from Rn to R
A function is a rule f that associates with each element in a set A one and only
one element in a set B.
If f associates the element b with the element a, then we write b = f(a) and say
that b is the image of a under f or that f(a) is the value of f at a.
The set A is called the domain of f and the set B is called the codomain of f.
The subset of B consisting of all possible values for f as a varies over A is
called the range of f.
Function from
Rn
to R m
If the domain of a function f is R n and the codomain is R m , then f is called a
m
map or transformation from R n to R . We say that the function f maps Rn
Into R m , and denoted by f : R n → R m .
If m = n the transformation f :
Rn → Rn is called an operator on Rn.
Suppose f1, f2, …, fm are real-valued functions of n real variables, say
w1 = f1(x1,x2,…,xn)
w2 = f2(x1,x2,…,xn)
…
wm = fm(x1,x2,…,xn)
These m equations assign a unique point (w1,w2,…,wm) in R m to each point
(x1,x2,…,xn) in R n and thus define a transformation from R n to R m . If we
m
n
denote this transformation by T: R → R then
T (x1,x2,…,xn) = (w1,w2,…,wm)
Example: The equations
w1  x1  x2
w2  3 x1 x2
w3  x12  x22
Defines a transformation T : R 2  R 3 .
With this transformation, the image of the point (x1, x2) is
T ( x1 , x2 )  ( x1  x2 ,3x1 x2 , x12  x22 )
Thus, for example, T(1, -2)=(-1, -6, -3)
Linear Transformations from
Rn
to R m
A linear transformation (or a linear operator if m = n) T:
defined by equations of the form
w1  a11 x1  a12 x2  ...  a1n xn
w2  a21 x1  a22 x2  ...  a2 n xn
...
wm  am1 x1  am 2 x2  ...  amn xn
or
R n→ R m
 w1   a11 a12
w  a
 2    21 a22
  
  
 wm   am1 am 2
is
a1n   x1 
a2 n   x2 
 
 
amn   xm 
or
w = Ax
The matrix A = [aij] is called the standard matrix for the linear transformation T,
and T is called multiplication by A.
Example: If the linear transformation T : R 4  R 3 is defined by the equations
w1  2 x1  3x2  x3  5 x4
w2  4 x1  x2  2 x3  x4
w3  5 x1  x2  4 x3
Find the standard matrix for T, and calculate T (1, 3, 0, 2)
Solution: T can be expressed as
 x1 
 w1   2 3 1 5  
 w    4 1 2 1   x2 
 2 
 x 
 w3   5 1 4 0   3 
 x4 
So the standard matrix for T is
 2 3 1 5
A   4 1 2 1 
5 1 4 0 
Furthermore, if ( x1 , x2 , x3 , x4 )  (1, 3,0, 2)
w1  2 x1  3x2  x3  5 x4  1
w2  4 x1  x2  2 x3  x4  3
w3  5 x1  x2  4 x3  8
Thus, T (1, 3, 0, 2)  (1,3,8)
Or
1 
 w1   2 3 1 5   1 
 w    4 1 2 1   3  3
 2 
 0   
 w3   5 1 4 0    8
2 
Remarks
Notations:
If it is important to emphasize that A is the standard matrix for T. We denote the linear
transformation T: R n → R m by TA: R n → R m . Thus, TA(x) = Ax
We can also denote the standard matrix for T by the symbol [T], or T(x) = [T]x
Remark:
We have establish a correspondence between m×n matrices and linear
transformations from R n to R m :
To each matrix A there corresponds a linear transformation TA (multiplication by A), and
to each linear transformation T: R n → R m , there corresponds an m×n matrix [T] (the
standard matrix for T).
Examples
m
n
Zero Transformation from R to R
If 0 is the m×n zero matrix and 0 is the zero vector in Rn, then for every
vector x in Rn
T0(x) = 0x = 0
n
So multiplication by zero maps every vector in R into the zero vector in
.R m . We call T0 the zero transformation from Rn to R m .
Identity operator on Rn
If I is the n×n identity, then for every vector in Rn
TI(x) = Ix = x
So multiplication by I maps every vector in Rn into itself. We call TI the
identity operator on Rn .
Reflection Operators
2
In general, operators on R and R 3 that map each vector into its symmetric
image about some line or plane are called reflection operators.
Such operators are linear.
Projection Operators
In general, a projection operator (or more precisely an orthogonal
projection operator) on R2 or R 3 is any operator that maps each
vector into its orthogonal projection on a line or plane through the origin.
The projection operators are linear.
Rotation Operators
2
An operator that rotate each vector in R through a fixed angle θ is called a
rotation operator on R2 .
Operator
Rotation
through an
angle 
Illustration
( w1 , w2 )

( x, y )
Equations
Standard
Matrix
w1  x cos   y sin 
cos 
 sin 

w2  x sin   y cos 
 sin  
cos  
Example: Use matrix multiplication to find the image of the vector (1, 1) when it is
rotated through an angle of 30 degree (  / 6 )
x 
x

Solution: the image of the vector
 y  is
 
cos  / 6  sin  / 6   x   3 / 2
w
  y  
sin

/
6
cos

/
6

    1/ 2
 3
1
x

1/ 2   x   2
2
   
3 / 2   y   1
3
x


2
2
So
 3/2
w
 1/ 2
 3 1
1/ 2  1  2 

   
3 / 2  1 1  3 


 2 

y


y

Dilation and Contraction Operators
2
If k is a nonnegative scalar, the operator on R or R 3 is called a contraction
with factor k if 0 ≤ k ≤ 1 and a dilation with factor k if k ≥ 1 .
Operator
Contraction
with factor k
on R 2
(0 ≤ k ≤ 1 )
Dilation with
factor k on R 2
(k ≥ 1 )
Illustrator
Equations
( x, y )
w1  kx
w2  ky
( kx, ky )
( kx, ky )
( x, y )
w1  kx
w2  ky
Standard
Matrix
k 0 
0 k 


k 0 
0 k 


Compositions of Linear Transformations
If TA : R n → R k and TB : R k → R m are linear transformations, then for
n
each x in R one can first compute TA(x), which is a vector in R k , and
then one can compute TB(TA(x)), which is a vector in R m .
Thus, the application of TA followed by TB produces a transformation from
to R m . This transformation is called the composition of TB with TA and is
denoted by TB TA . Thus (TB TA )(x)  (TB (TA (x))
The composition TB TA
is linear since
(TB TA )(x)  (TB (TA (x))  B( Ax)  ( BA)x
The standard matrix for TB TA is BA. That is, TB TA  TBA
Rn
Remark:
TB TA  TBA captures an important idea: Multiplying matrices is equivalent to
composing the corresponding linear transformations in the right-to-left order of
the factors.
n
k
k
m
Alternatively, If T1 : R  R and T2 : R  R are linear transformations, then because
the standard matrix for the composition T2 T1 is the product of the standard matrices
of T2 and T1, we have
T2
T1   T2  T1 
Example: Find the standard matrix for the linear operator T : R 2  R 2 that first reflects
A vector about the y-axis, then reflects the resulting vector about the x-axis.
Solution: The linear transformation T can be expressed as the composition
T  T2 T1
Where T1 is the reflection about the y-axis, and T2 is the reflection about
The x-axis.
 1 0
,
1


T1    0
Sine the standard matrix for T is
1
T2   0

T   T2
0
1
T1   T2  T1 
1 0   1 0  1 0 
T

  0 1  0 1   0 1


 

Which is called the reflection about the origin.
Note: the composition is NOT commutative.
Example: Let T1 : R 2  R 2 be the reflection operator about the line y=x, and let
T2 : R 2  R 2 be the orthogonal projection on the y-axis. Then
T1
0 1  0 0  0 1 
T2   T1  T2   
 0 1   0 0 
1
0


 

T2
0 0  0 1  0 0 
T1   T2  T1   
 1 0   1 0 
0
1


 

T2
T1   T1 T2 
Thus, T2 T1 and T2 T1 have different effects on a vector x.
4.3 Properties of Linear Transformations form Rn to R m
One-to-One Linear transformations
Definition
n
m
A linear transformation T : R  R
is said to be one-to-one if T maps
n
distinct vectors (points) in R into distinct vectors (points) in R m
Remark:
That is, for each vector w in the range of a one-to-one linear transformation
T, there is exactly one vector x such that T(x) = w.
Example:
2
2
The linear operator T: R  R that rotates each vector through an angle
a one-to-one linear transformation.
 is
In contrast, if T: R 2  R 2 is the orthogonal projection on the x-axis, then it’s
not a one-to-one linear transformation.
Theorem 4.3.1 (Equivalent Statements)
If A is an n×n matrix and TA : R  R is multiplication by A, then the
following statements are equivalent.
n
•A is invertible
n
•The range of TA is R
•TA is one-to-one
n
Example
The rotation operator T : R 2  R 2 that rotates each vector through an angle 
is one-to-one.
cos   sin  
T



The standard matrix for T is
 sin  cos  


which is invertible since
det[T ] 
cos 
sin 
 sin 
 cos 2   sin 2   1  0
cos 
Example
If T: R 2  R 2 is the orthogonal projection on the x-axis, then it’s
not one-to-one.
1 0 
T

The standard matrix for T is   

0 0 
Which is not invertible since det[T] = 0
Inverse of a One-to-One Linear Operator
Suppose TA : R  R is a one-to-one linear operator
⇒ The matrix A is invertible.
⇒ TA-1 : R n  R n is itself a linear operator; it is called the inverse of TA.
⇒ T (T 1 (x))  AA1x  Ix  x
n
A
n
A
TA1 (TA (x))  A1 Ax  Ix  x
⇒
TA TA1  TAA1  TI
TA1 TA  TA1 A  TI
If w is the image of x under TA, then TA-1 maps w back into x, since
TA1 (w)  TA1 (TA (x))  x
When a one-to-one linear operator on R n is written as T : R n  R n , then
the inverse of the operator T is denoted by T 1 .
1
Thus, by the standard matrix, we have T   T 
1
Example
2
2
Show that the linear operator T : R  R
w1= 2x1+ x2
w2 = 3x1+ 4x2
1
is one-to-one, and find T ( w1 , w2 )
defined by the equations
 w1   2 1   x1 


 x 
w
3
4
 2
 2 
Solution: The matrix form of these equations is 
2 1

3
4


So the standard matrix for T is [T ] 

This matrix is invertible and the standard matrix for T 1 is
1
 4

 5
5
1
1
[T ]  [T ]  

 3 2 
 5 5 
Thus
1
1
 4
4


w

w
1
2
 5

w
5   w1   5
5
1  1 
[T ]    
   

w
3
2
w
3
2
 2  
  2   w  w 
 5 5 
 5 1 5 2 
from which we conclude that
4
1
3
2
T 1 ( w1 , w2 )  ( w1  w2 ,  w1  w2 )
5
5
5
5
Linearity Properties
Theorem 4.3.2 (Properties of Linear Transformations)
A transformation T : R n  R m is linear if and only if the following
relationships hold for all vectors u and v in Rn and every scalar c.
T(u + v) = T(u) + T(v)
T(cu) = cT(u)
Example: Determine whether T: R 2  R 2 is a linear operator if T(x, y)=(x, 3y).
Solution:
T (( x1 , y1 )  ( x2 , y2 ))  T ( x1  x2 , y1  y2 )  ( x1  x2 ,3( y1  y2 ))
 ( x1 ,3 y1 )  ( x2 ,3 y2 )  T ( x1 , y1 )  T ( x2 , y2 )
T (c( x, y ))  T (cx, cy )  (cx,3cy)  c( x,3 y)  cT ( x, y)
So T(x, y)= (x, 3y) is a linear operator.
We call the vectors e1, e2, …, en be the standard basis vectors for Rn if
1 
0
0 
1 
 
 
e1  0  , e 2  0  ,
 
 
 
 
0
0
0
0
 
, e n  0 
 
 
1 
In particular, In R2 and R 3 the standard basis vectors are the vectors of length 1
Along the coordinate axes.
Theorem 4.3.3
If T : R n  R m is a linear transformation, and e1, e2, …, en are the
standard basis vectors for Rn , then the standard matrix for T is
A = [T] = [T(e1) | T(e2) | … | T(en)]
Example:
2
2
Find the standard matrix for T: R  R from the images of the standard
Basis vectors if T dilates a vector by a factor of 2, then reflects that vector about the
line y=x, and then projects that vector orthogonally onto x-axis.
Solution: Here
e1  (1, 0)  (2, 0)  (0, 2)  (0, 0)  0e1  0e2
e2  (0,1)  (0, 2)  (2, 0)  (2, 0)  2e1  0e2
Thus
0 2 
[T ]  

0 0 
Eigenvalue and Eigenvector
Definition
n
n
If T: R  R is a linear operator, then a scalar λ is called an eigenvalue of T
n
if there is a nonzero x in Rsuch
that
T(x) = λx.
Those nonzero vectors x that satisfy this equation are called the
eigenvectors of T corresponding to λ
Remarks:
If A is the standard matrix for T, then the equation becomes Ax = λx
•
•
The eigenvalues of T are precisely the eigenvalues of its standard matrix A
x is an eigenvector of T corresponding to λ if and only if x is an eigenvector of A
corresponding to λ
If λ is an eigenvalue of A and x is a corresponding eigenvector, then Ax = λx, so
multiplication by A maps x into a scalar multiple of itself
2
3
In R and R, this means that multiplication by A maps each eigenvector x into a
Vector x intro a vector that lies on the same line as x.
x
Ax   x
x
 0
Ax   x
 0
Example:
Let T: R 2  R 2 be the reflection about the y-axis. Find the eigenvalues and
corresponding eigenvectors of T. Check your calculations By calculating the
eigenvalues and corresponding eigenvectors from the standard matrix for T.
Solution: This transformation maps vectors on the x-axis to their negatives,
vectors on the y-axis into themselves, and maps no other vectors into
scalar multiples of themselves.
Thus the eigenvalues are λ = ±1 and the eigenvectors are vectors (x, y)
with either x = 0 or y = 0, but not both. To verify this, we observe that since
e1 → -e1 and e2 → e2, the standard matrix for the transformation is
 1 0
 0 1


. Hence the characteristic equation is
1 0   1 0 
det( 

)0


0 1   0 1 
2
or   1  0 with solutions λ = ±1.
If (x, y) is an eigenvector corresponding to λ = 1,
Then  2 0   x  0 
0 0  y   0

   
 2 x  0
0   0
   
or x = 0, so the vector must lie on the y-axis.
If (x, y) is an eigenvector corresponding to λ = –1, then
0 0   x  0
0 2   y   0 

   
0  0
 2 y   0 

  
or y = 0, so the vector must lie on the x-axis.
Theorem 4.3.4 (Equivalent Statements)
If A is an n×n matrix, and if TA : R n  R n is multiplication by A, then the following
are equivalent.
• A is invertible
• Ax = 0 has only the trivial solution
• The reduced row-echelon form of A is In
• A is expressible as a product of elementary matrices
• Ax = b is consistent for every n×1 matrix b
• Ax = b has exactly one solution for every n×1 matrix b
• det(A) ≠ 0
• The range of TA is Rn
• TA is one-to-one