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
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
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
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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