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Question: 31 please...

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28. Prove that if W - Span( wi, w2,..., Wi) 3 R, then: W {VERIyow = 0 for all i=1 k}. Hint: start by writing the complete (original) definition of Wi for any subspace W ofR 29. Prove that in any IR: (a) o), and (b) (R). Hint: for (a) write the definition of fo definition, and for (b) use the previous Exercise and the fact that R is Spanned by e through e. and explain why every vector in R satisfies this 30. Prove The Dimension Theorem for Orthogonal Complements: If W is a subspace of R with orthogonal complement Wa, then: dim(W) + dim(W) n. Hint: Assemble a basis for W in tho rows of a matrix and apply The Dimension Theorem for Matrices. 31. Let W s R. Prove that Wnwoy. Hint: Suppose v e W and W e W. What can you 32. Use the idea behind the previous Exercise to give another proof that (R)0 33. Let Ws R. Our goal in this Exercise is to prove that (W) W. say about the dot product of w with itself? in why Exercise 29 takes care of the cases when W is one of the trivial subspaces. For the rest of this Exercise, we can therefore assume that Wis a non-trivial subspace. b. Let us use the symbol U for Wa. Prove that W c U. Hint: write down the definition of U (which is W4) as well as U. Stare at these two definitions until you can clearly explain in writing why every vector in Walso satisfies the definition of U c. In Exercise 39 of Section 1.7, we discussed the concept of nested subspaces. Part (b) tells us that WSUR is a nesting of subspaces. Use The Dimension Theorem fo l Comnlements as well as Exercise 39 of Section 1.7 to show that in fact 31 please
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