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Unformatted text preview: Math 167 homework 5 solutions November 4, 2010 3.3.4: Write out E 2 = || Ax- b || 2 and set to zero its derivatives with respect to u and v , if A = 1 0 0 1 1 1 , x = bracketleftbigg u v bracketrightbigg , b = 1 3 4 . Compare the resulting equations with A T A x = A T b , confirming that calculus as well as geometry gives the normal equations. Find the solutions x and the projection p = A x . Why is p = b ? Note that u ( Ax- b ) = 1 1 and v ( Ax- b ) = 1 1 . Then the conditions 0 = u ( E 2 ) = 2 parenleftbigg u ( Ax- b ) parenrightbigg T ( Ax- b ) and 0 = v ( E 2 ) = 2 parenleftbigg v ( Ax- b ) parenrightbigg T ( Ax- b ) imply that bracketleftbigg 2 1 1 2 bracketrightbigg x = bracketleftbigg 5 7 bracketrightbigg . Solving this linear system gives u = 1 and v = 3. We then check that we have a minimum by taking the Hessian of E 2 , or in this case just check that the 1 matrix of the linear system above is positive definite. Note that the linear system above is just the normal equations. Finally, p = A x = 1 3 4 ....
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