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HW5_solutions - Math 167 homework 5 solutions November 4...

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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 ˆ x = A T b , confirming that calculus as well as geometry gives the normal equations. Find the solutions ˆ x and the projection p = A ˆ x . Why is p = b ? Note that ∂u ( Ax - b ) = 1 0 1 and ∂v ( Ax - b ) = 0 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
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matrix of the linear system above is positive definite. Note that the linear system above is just the normal equations. Finally, p = A ˆ x = 1 3 4 . Thus p = b and so b must be in the column space of A . Equivalently, E 2 = 0 for the least-squares solution ˆ x .
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