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m425b-hw5soln-s08

m425b-hw5soln-s08 - MATH 425b ASSIGNMENT 5 SOLUTIONS SPRING...

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MATH 425b ASSIGNMENT 5 SOLUTIONS SPRING 2008 Prof. Alexander Chapter 9: (3) If x 6 = y then Ax - Ay = A ( x - y ) 6 = 0, that is, Ax 6 = Ay . This shows A is one-to-one. (4) Let T L ( X, Y ), and let z Z = range( T ) = { y Y : y = Tx for some x X } . We show Z is a vector space: suppose u, v Z and c is a scalar. Then u = Tx, v = Tw for some x, w X , and T ( cx + w ) = cTx + Tw = cu + v , so cu + v Z . Thus Z is a vector space. Next let N = { x X : Tx = 0 } be the nullspace. Suppose g, h N and c is a scalar. Then T ( cg + h ) = cTg + Th = c · 0 + 0 = 0, so cg + h N . Thus N is a vector space. (5) Let A L ( R n , R ), let { e 1 , .., e n } be the standard basis, let y i = Ae i R , and let y = ( y 1 , .., y n ). The for every x R n , A x = A ( n X i =1 x i e i ) = n X i =1 x i Ae i = n X i =1 x i y i = x · y . By the Schwarz inequality, for all x 6 = 0, | A x | = | x · y | ≤ | x | | y | , so sup x 6 =0 | A x | | x | ≤ | y | , meaning || A || ≤ | y | . To show equality actually holds, consider two cases: if y = 0, then A = 0, so || A || = | y | . If y 6 = 0 then || A || = sup x 6 =0 | A x | | x | | A y | | y | = | y | 2 | y | = | y | , so again || A || = | y | .

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