Theorem. (The Gram-Schmidt Process)
Given a basis
for a subspace W of
Rn, define
Then
is an orthogonal basis for W and
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Example. Find an orthonormal basis for the column
space of A:
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Using earlier calculations, find a QR factorization of A.
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6.5 Least-Squares Problems
If A is mxn and b is in Rm, a least-squares solution
of Ax=b is an in Rn such that
for all x in Rn.
Theorem. The set of least squares solutions of Ax=b
is the set of all solutions of
(normal equations)
Least-squares error =
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When is the least-squares solution unique?
Theorem. The matrix
is invertible if and only if
the columns of A are linearly independent. In this
case, the equation
has only one leastsquares solution
and it is given by
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Example. Find the least-squares solution of the
inconsistent system
where
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Calculating Least-Squares Solutions using the
QR Factorization:
Theorem. Given an mxn matrix of rank n, let A=QR be
a QR factorization of A. Then, for each in Rm, the
equation
has a unique least squares
solution, given by
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Calculating Least-Squares Solutions using the
QR Factorization:
Theorem. Given an mxn matrix of rank n, let A=QR be
a QR factorization of A. Then, for each in Rm, the
equation
has a unique least squares
solution, given by
(The solution may be obtained by using back
substitution to solve
)
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Example. Using a QR factorization, find the leastsquares solution to the inconsistent system
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6.6 Applications to Linear Models
Least-Squares Lines:
Given a table of data
we wish to find a linear function
that best fits the data in the least squares sense.
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Example. Find an equation of the form
that best fits the given data points:
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Least-Squares Fitting of other Curves
If the data do not resemble a linear function, we could
use a higher degree polynomial.
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Least-Squares Fitting of other Curves
If the data do not resemble a linear function, we could
use a higher degree polynomial.
To find the best least-squares fit to the data
by a polynomial of the form
we must find the least-squares solution to the system
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Example. Find the best quadratic least-squares fit to the data
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Course Evaluations..
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