5.1%20Length%20and%20Dot%20Product%20in%20Rn

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5.1 Length and Dot Product in Rn
Chapter 5 Inner Product Spaces

• Notes The length of a vector is also called its
  norm.

• Notes

is called a unit vector.
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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

• Ex
• the standard unit vector in R2
• the standard unit vector in R3

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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

(A vector
in Rn is represented as an n1 column matrix)
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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

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• Note

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• Properties of norm
• (1)
• (2) if and only if
• (3)

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• Properties of distance
• (1)
• (2) if and only if
• (3)

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• Note
• If v is a init vector, then
  .
• The formula for the orthogonal projection of u
  onto v takes the following simpler form.

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5.3 Orthonormal Bases Gram-Schmidt Process

• 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.

• Notes

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• Notes

• Ex

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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
• 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
• (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
  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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• Note
• The problem of finding the least squares solution
  of
• is equal to he problem of finding an exact
  solution of the
• associated normal system .

• Thm
• For any linear system , the
  associated normal system
• 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

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• Thm
• If A is an mn matrix with linearly independent
  column vectors, then for every m1 matrix b, the
  linear system Ax b has a unique least squares
  solution. This solution is given by
• Moreover, if W is the column space of A, then the
  orthogonal projection of b on W is

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5.5 Applications of Inner Product Spaces
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• Note Ca, b is the inner product space of all
  continuous
• functions on a, b.

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