E7+Discussion+2010-08-02

E7+Discussion+2010-08-02 - Monday, August 2, 2010 E7...

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Monday, August 2, 2010 E7 DISCUSSION SECTION
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Outline Matrix equations Rank of a matrix Linear systems of equations Least squares approximation Linear Regression Interpolation Numerical Integration Trapezoidal Rule Simpson’s Rule Adaptive Implementation
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Rank of a Matrix Think of a matrix in terms of its columns Each a i is m -by-1 vector and A set of vectors is linearly independent if none of them can be written as a linear combination of the others: The column rank of A is the number of linearly independent columns The row rank is the number of linearly independent rows Column rank and row rank are always equal
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Systems of Linear Equations We’ve seen many problems of the form Existence of a solution: Can we write b as a linear combination of a 1 , a 2 , …, a n ? Is b linearly dependent on { a 1 , a 2 , … , a n }? Are there the same number of linearly independent vectors in { a 1 , a 2 , … , a n , b } as in { a 1 , a 2 , … , a n }? Does rank([ A b ]) equal rank( A )? Rank gives us an easy way to check for existence
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Systems of Linear Equations We’ve seen many problems of the form Uniqueness of a solution: Are there as many equations as unknowns? Are there n linearly independent rows in [ A b ]? Does rank([ A b ]) equal n ? Again, rank gives a simple test for uniqueness
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Least Squares Approximation If we can’t solve Ax = b, then we will have error We want to make the error “small” by minimizing the Euclidean norm There are many norms, which all satisfy minimize the sum of squared errors
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Uniqueness of Minima Suppose c
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This note was uploaded on 08/14/2010 for the course E 7 taught by Professor Patzek during the Summer '08 term at University of California, Berkeley.

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E7+Discussion+2010-08-02 - Monday, August 2, 2010 E7...

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