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Chapter 5
Inner Product Spaces
n
5.1 Length and Dot Product in R

Notes: The length of a vector is also called its norm.

Notes:
1 
v 0
2 
v 1 v
3 
v  0 iff
is called a unit vector.
v 0
5-1
5-2
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• Notes:
The process of finding the unit vector in the direction of v is
called normalizing the vector v.
• A standard unit vector in Rn:
e 1 , e 2 ,  , e n   1,0 ,  ,0 , 0 ,1,  ,0 , 0 ,0 ,  ,1

Ex:
the standard unit vector in R2: i , j   1, 0 , 0 ,1 
the standard unit vector in R3:
i ,
j, k  
1, 0 , 0 , 0 ,1, 0 , 0 , 0 ,1 
5-4

Notes: (Properties of distance)
(1)
d (u , v )  0
(2)
d (u , v )  0
(3)
d (u , v )  d ( v , u )
if and only if
u  v
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5-6
• Euclidean n-space:
Rn was defined to be the set of all order n-tuples of real
numbers. When Rn is combined with the standard
operations of vector addition, scalar multiplication,
vector length, and the dot product, the resulting vector
space is called Euclidean n-space.
5-7

Dot product and matrix multiplication:
 u1 
 
u2


u
  
 
u n 
 v1 
 
v2


v
  
 
v n 
u  v  u v  [u 1
T
u2
(A vector u  ( u1 , u 2 ,  , u n ) in Rn
is represented as an n×1 column matrix)
 v1 
 
v2

  [u v  u v    u v ]
 un ]
1 1
2 2
n n
 
 
v n 
5-8

Note: The angle between the zero vector and another vector
is not defined.
5-9

Note: The vector 0 is said to be orthogonal to every vector.
5-10

Note:
Equality occurs in the triangle inequality if and only if
the vectors u and v have the same direction.
5-11
5.2 Inner Product Spaces
• Note:
u  v  dot product ( Euclidean
n
inner product for R )
 u , v  general inner product for vector
space V
5-12

Note:
A vector space V with an inner product is called an inner
product space.
Vector space:
V ,
, 
Inner product space:
V ,
 , ,  ,  
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
Note: || u || 2 〈 u , u 〉
5-15

Properties of norm:
(1) || u ||  0
(2) || u ||  0 if and only if
u0
(3) || c u ||  | c | || u ||
5-16

Properties of distance:
(1) d ( u , v )  0
(2) d ( u , v )  0 if and only if u  v
(3) d ( u , v )  d ( v , u )
5-17

Note:
If v is a init vector, then 〈 v , v 〉 || v || 2  1.
The formula for the orthogonal projection of u onto v
takes the following simpler form.
proj v u   u , v  v
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5.3 Orthonormal Bases: Gram-Schmidt Process

S  v 1 , v 2 ,  , v n   V
S  v 1 , v 2 ,  , v n   V
vi, v j  0
1
vi, v j  
0
i j
i j
Note:
If S is a basis, then it is called an orthogonal basis or an
orthonormal basis.
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5.4 Mathematical Models and Least Squares
Analysis
5-26

Orthogonal complement of W:
Let W be a subspace of an inner product space V.
(a) A vector u in V is said to orthogonal to W,
if u is orthogonal to every vector in W.
(b) The set of all vectors in V that are orthogonal to W is
called the orthogonal complement of W.
W

 {v  V |  v , w   0 ,  w  W }
W
 Notes:
(1)
0 
V

(read “ W perp”)
(2) V

 0 
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• Notes:
W is a subspace
(1) W

is a subspace
(2) W  W
(3)
of V


of V
 0 

(W )  W
 Ex:
If V  R , W  x  axis
2
Then (1) W

 y - axis
(2) W  W
(3) (W


is a subspace
of R
2
 ( 0 , 0 )

) W
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• Notes:
(1) Among all the scalar multiples of a vector u, the
orthogonal projection of v onto u is the one that is
closest to v.
(2) Among all the vectors in the subspace W, the vector
proj W v is the closest vector to v.
5-32
• The four fundamental subspaces of the matrix A:
N(A): nullspace of A
N(AT): nullspace of AT
R(A): column space of A
R(AT): column space of AT
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
Least squares problem:
Ax  b
m  n n 1 m 1
(A system of linear equations)
(1) When the system is consistent, we can use the Gaussian
elimination with back-substitution to solve for x
(2) When the system is inconsistent, how to find the “best
possible” solution of the system. That is, the value of x for
which the difference between Ax and b is small.

Least squares solution:
Given a system Ax = b of m linear equations in n unknowns,
the least squares problem is to find a vector x in Rn that
minimizes
Ax  b
with respect to the Euclidean inner
product on Rn. Such a vector is called a least squares
solution of Ax = b.
5-35
A M
x R
mn
n
A x  CS ( A ) ( CS  A  is a subspace
m
of R )
W  CS ( A )
Let A xˆ  proj W b
 ( b  A xˆ )  CS ( A )
 b  A xˆ  ( CS ( A ))


 NS ( A )

 A ( b  A xˆ )  0
i.e.


A A xˆ  A b
(the normal equations of the least squares
problem Ax = b)
5-36
• Note:
The problem of finding the least squares solution of A x  b
is equal to he problem of finding an exact solution of the
associated normal system A  A xˆ  A  b .

Thm:
For any linear system A x  b , the associated normal system


A A xˆ  A b
is consistent, and all solutions of the normal system are least
squares solution of Ax = b. Moreover, if W is the column space
of A, and x is any least squares solution of Ax = b, then the
orthogonal projection of b on W is
proj W b  A x
5-37
• Thm:
If A is an m×n matrix with linearly independent column vectors,
then for every m×1 matrix b, the linear system Ax = b has a
unique least squares solution. This solution is given by

x  ( A A)
1

A b
Moreover, if W is the column space of A, then the orthogonal
projection of b on W is

proj W b  A x  A ( A A )
1

A b
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5.5 Applications of Inner Product Spaces
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• Note: C[a, b] is the inner product space of all continuous
functions on [a, b].
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