Transcript 4.4 ppt File

4 Vector Spaces
4.4
COORDINATE SYSTEMS
© 2012 Pearson Education, Inc.
THE UNIQUE REPRESENTATION THEOREM

Theorem 7: Let B  {b1 ,...,b n } be a basis for
vector space V. Then for each x in V, there exists a
unique set of scalars c1, …, cn such that
----(1)
x  c1b1  ...  cn bn

Proof: Since B spans V, there exist scalars such that
(1) holds.
Suppose x also has the representation

x  d1b1  ...  d n b n
for scalars d1, …, dn.
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Slide 4.4- 2
THE UNIQUE REPRESENTATION THEOREM
 Then, subtracting, we have
0  x  x  (c1  d1 )b1  ...  (cn  d n )b n ----(2)
 Since B is linearly independent, the weights in (2)
must all be zero. That is, c j  d j for 1  j  n .
 Definition: Suppose B  {b1 ,...,b n } is a basis for V
and x is in V. The coordinates of x relative to the
basis B (or the B-coordinate of x) are the weights
c1, …, cn such that x  c1b1  ...  cn b n .
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Slide 4.4- 3
THE UNIQUE REPRESENTATION THEOREM
 If c1, …, cn are the B-coordinates of x, then the vector
n
 c1 
in
[x]B   
 
cn 
is the coordinate vector of x (relative to B), or the
B-coordinate vector of x.
 The mapping x  x  B is the coordinate mapping
(determined by B).
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Slide 4.4- 4
COORDINATES IN
n
 When a basis B for
is fixed, the B-coordinate
vector of a specified x is easily found, as in the
example below.
2
 1
4
 Example 1: Let b1    , b 2    , x    , and
n
 1
 1
 5
B  {b1 , b 2 }. Find the coordinate vector [x]B of x
relative to B.
 Solution: The B-coordinate c1, c2 of x satisfy
2
 1  4 
c1    c2     
 1
 1  5
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b1
b2
x
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COORDINATES IN
or
n
 2 1  c1   4 
 1 1  c    5

 2  
b1
b2
----(3)
x
 This equation can be solved by row operations on an
augmented matrix or by using the inverse of the
matrix on the left.
 In any case, the solution is c1  3 , c2  2 .
 Thus x  3b1  2b 2 and
 c1   3 .
x
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B
  
c2   2 
Slide 4.4- 6
COORDINATES IN
n
 See the following figure.
 The matrix in (3) changes the B-coordinates of a
vector x into the standard coordinates for x.
 An analogous change of coordinates can be carried
n
out in
for a basis B  {b1 ,...,b n }.
 Let PB
  b1 b2
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bn 
Slide 4.4- 7
COORDINATES IN
n
 Then the vector equation
is equivalent to
x  c1b1  c2 b2  ...  cn b n
x  PB  x B
----(4)
 PB is called the change-of-coordinates
matrix
n
from B to the standard basis in
.
 Left-multiplication by PB transforms the coordinate
vector [x]B into x.
 Since the columns of PB form a basis for
, PB is
invertible (by the Invertible Matrix Theorem).
n
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Slide 4.4- 8
COORDINATES IN
n
1
B
 Left-multiplication by P converts x into its Bcoordinate vector:
P x   x B
1
B
 The correspondence x
1
x
 B, produced by PB , is
the coordinate mapping.
1
 Since PB is an invertible matrix, the coordinate
mapping is a one-to-one linear transformation from
n
onto
, by the Invertible Matrix Theorem.
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n
Slide 4.4- 9
THE COORDINATE MAPPING
 Theorem 8: Let B  {b1 ,...,b n } be a basis for a vector
 x nB is a
space V. Then the coordinate mapping x
one-to-one linear transformation from V onto .
 Proof: Take two typical vectors in V, say,
u  c1b1  ...  cn b n
w  d1b1  ...  d n b n
 Then, using vector operations,
u  v  (c1  d1 )b1  ...  (cn  d n )b n
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Slide 4.4- 10
THE COORDINATE MAPPING
 It follows that
u  w 
B
 c1  d1   c1   d1 









  u B   w B

    
cn  d n  cn   d n 
 So the coordinate mapping preserves addition.
 If r is any scalar, then
ru  r (c1b1  ...  cn b n )  (rc1 )b1  ...  (rcn )b n
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Slide 4.4- 11
THE COORDINATE MAPPING
 So
 ru 
B
 rc1 
 c1 
    r    r  u B
 
 
 rcn 
 cn 
 Thus the coordinate mapping also preserves scalar
multiplication and hence is a linear transformation.
 The linearity of the coordinate mapping extends to
linear combinations.
 If u1, …, up are in V and if c1, …, cp are scalars, then
c1u1  ...  c p u p   c1  u1 B  ...  c p  u p  ----(5)
B
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B
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THE COORDINATE MAPPING
 In words, (5) says that the B-coordinate vector of a
linear combination of u1, …, up is the same linear
combination of their coordinate vectors.
 The coordinate mapping in Theorem 8 is an important
n
example of an isomorphism from V onto .
 In general, a one-to-one linear transformation from a
vector space V onto a vector space W is called an
isomorphism from V onto W.
 The notation and terminology for V and W may differ,
but the two spaces are indistinguishable as vector
spaces.
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Slide 4.4- 13
THE COORDINATE MAPPING
 Every vector space calculation in V is accurately
reproduced in W, and vice versa.
 In particular, any real vector space with a basis of n
vectors is indistinguishable from n .
 3
 1
 3
 Example 2: Let v1   6  , v 2   0  , x  12  ,
 
 
 
 2 
 1
 7 
and B  {v1 , v 2 }. Then B is a basis for
H  Span{v1 ,v 2 } . Determine if x is in H, and if it is,
find the coordinate vector of x relative to B.
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Slide 4.4- 14
THE COORDINATE MAPPING
 Solution: If x is in H, then the following vector
equation is consistent:
 3
 1  3
c1  6   c2  0   12 
 
   
 2 
 1  7 
 The scalars c1 and c2, if they exist, are the Bcoordinates of x.
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Slide 4.4- 15
THE COORDINATE MAPPING
 Using row operations, we obtain
 3 1 3 
 6 0 12 


 2 1 7 
 1 0 2
 0 1 3 .


0 0 0 
2
 Thus c1  2 , c2  3 and [x]B    .
 3
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Slide 4.4- 16
THE COORDINATE MAPPING
 The coordinate system on H determined by B is
shown in the following figure.
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Slide 4.4- 17