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Unformatted text preview: Partial Key to the Final Exam, Linear Algebra, Spring 2003 (The questions will be posted separately) 1) Proofs about ”orthogonal” or ”symmetric” etc are usually easy, because these words can be defined by equations. Which means the proof is mainly a calculation. In this problem, do not talk about perpendicular vectors etc. Instead: Answer: If A and B are orthogonal, we know that A T A = I and B T B = I . We must check this for AB . From matrix algebra and the assumptions, we get ( AB ) T AB = B T A T AB = B T IB = I , as desired. 2) Do the usual 6.1 calculations. Get λ = 1 or 3 and put the evecs into a matrix X . Then normalize the columns to get: U = 2 1 / 2 1 1 1 1 3) We must use the definition of subspace here. 1) [closure of scalarmult] Assume v ∈ U ∩ V (so, it is in U and in V ) and that α ∈ R is a scalar. Since v ∈ U and U is a subspace, we know α v ∈ U . Likewise, we can show α v ∈ V , so α v ∈ U ∩ V ....
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Vector Space, Euclidean vector, simple deﬁning formulas

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