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HW06 - of just linear operators as given in the deﬁnition...

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EE 203000 Linear Algebra Homework #6 (Due January 13, 2010 BEFORE noon to TA in Rm 606) Note: Detailed derivations are required to obtain a full score for each problem. (Total 120%) 1. (10%) Prove Theorem 6.1(e) and use this result to show that, if h x, z i = 0 for all x V , then z = 0. 2. (6%+6%) Let S be a subset of inner product space V . For the following cases, find (i) an orthogonal basis for span( S ) using Gram-Schmidt process; (ii) an orthonormal basis β ; (iii) the expression for x as a linear combination of vectors in β . (a) V M 2 × 2 ( R ), S = 3 5 - 1 1 ! , ˆ - 1 9 4 - 1 ! , ˆ 7 - 16 2 - 6 !) and x = ˆ - 1 26 - 4 8 ! . (b) V C 4 , S = { (1 , i, 2 - i, - 2) , (2 + 3 i, 3 i, 1 - i, 2 i ) , ( - 1 + 7 i, 6 + 10 i, 13 - 4 i, 3 + 4 i ) } and x = ( - 2 + 7 i, 6 + 9 i, 9 - 3 i, 4 + 4 i ). 3. (8%) Prove Parseval’s identity in Problem 15(a) of Section 6.2. 4. (8%) Problem 8 of Section 6.3. 5. (8%+8%) The concept of an adjoint can be extended to linear transformations (instead
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Unformatted text preview: of just linear operators) as given in the deﬁnition on page 367. Please answer the following questions regarding adjoints of linear transformations. Let T : V → W be a linear transformation, where V and W are ﬁnite-dimensional inner product spaces with inner products h· , ·i 1 and h· , ·i 2 , respectively. (a) Prove that [ T * ] β γ = ([ T ] γ β ) * , where β and γ are orthonormal bases for V and W , respectively. (b) Prove that that the null space of T is equal to the orthogonal subspace of the range space of T * , i.e., ( R ( T * )) ⊥ = N ( T ). 6. (24%) Problem 7 of Section 6.4 7. (6%+6%) Problem 17 (a), (b) of Section 6.4. 8. (10%) Problem 10 of Section 6.5 9. (20%) Problem 29 of Section 6.5. 1...
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