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Unformatted text preview: Homework 12 - Solutions 6.2 Problem 7 z W if and only if h z, v i = 0 for all v W . Writing v as a linear combination of vectors from , we see that this is equivalent to h z, X j j v j i = 0 This happens if and only if h z, v j i = 0 for all j . 6.2 Problem 11 The ij th etry of AA is the inner product of the j th and i th rows od A . Therefore, AA = I if and only if h v j , v k i = ij whete ij is the Kronecker delta (See page 89). this proves that the rows form and ONB of C n . 6.2 Problem 13 (a) Let x S . Then h x, y i = 0 for all y S , since S S . Therefore, S S . (b) Let x S . For any y S , h x, y i = 0. Therefore, y ( S ) , proving S ( S ) . Since S is a subspace even if S is not, taking the span on both side of the above equation, the proof is complete. (c) By (b), W ( W ) . Suppose x ( W ) . Then, for any z W , h x, z i = 0. On the other hand if x /...
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- Spring '10