Transcript Document
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:
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v 0
v 1 v is called a unit vector.
v 0 iff v 0
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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:
e1 , e2 ,, en 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
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Notes: (Properties of distance)
(1)
d (u , v) 0
(2)
d (u , v) 0 if and only if
(3)
d (u , v) d ( v , u)
uv
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• 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.
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Dot product and matrix multiplication:
u1
u
u 2
u n
v1
v
v 2
v n
u v u T v [u1 u 2
(A vector u (u1 , u2 , , un ) in Rn
is represented as an n×1 column matrix)
v1
v
u n ] 2 [u1v1 u 2 v 2 u n v n ]
v n
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Note: The angle between the zero vector and another vector
is not defined.
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Note: The vector 0 is said to be orthogonal to every vector.
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Note:
Equality occurs in the triangle inequality if and only if
the vectors u and v have the same direction.
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5.2 Inner Product Spaces
• Note:
u v dot product (Euclidean inner product for R n )
u , v general inner product for vector space V
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Note:
A vector space V with an inner product is called an inner
product space.
Vector space:
Inner product space:
V ,
V ,
,
, , ,
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Note: || u ||2 〈u , u〉
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Properties of norm:
(1) || u || 0
(2) || u || 0 if and only if u 0
(3) || cu || | c | || u ||
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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)
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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.
projv u u , v v
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5.3 Orthonormal Bases: Gram-Schmidt Process
S v1 , v 2 ,, v n V
S v1 , 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
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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 (read “ W perp”)
Notes:
(1)
0 V
(2) V 0
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• Notes:
W is a subspace of V
(1) W is a subspace of V
(2) W W 0
(3) (W ) W
Ex:
If V R 2 , W x axis
Then (1) W y - axis is a subspace of R 2
(2) W W (0,0)
(3) (W ) 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
projW v is the closest vector to v.
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• 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.
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A M mn
x Rn
Ax CS ( A) (CS A is a subspace of R m )
W CS ( A)
Let Axˆ projW b
(b Axˆ ) CS ( A)
b Axˆ (CS ( A)) NS ( A )
A (b Axˆ ) 0
i.e.
A Axˆ Ab (the normal equations of the least squares
problem Ax = b)
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• Note:
The problem of finding the least squares solution of Ax b
is equal to he problem of finding an exact solution of the
associated normal system A Axˆ A b .
Thm:
For any linear system Ax b , the associated normal system
A Axˆ 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
projW b Ax
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• 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 Ab
Moreover, if W is the column space of A, then the orthogonal
projection of b on W is
projW b Ax A( A A) 1 Ab
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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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