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Question: 31...

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28, Prove that if W-Span( { w , w2,、.., ws) )ヨR, then: W0 forall - I..k) Hint: start by writing the complete (original) definition of W for any subspace WofR 29. Prove that in any R )nd (b)(R. Hint: for (a) write the definition of 0.) and explain why every vector in R satisfies tis definition, and for (b) use the previous Exercise and the fact that R is Spanned by e, throughe 30. Prove The Dimension Theorem for Orthogonal Complements: If W is a subspace of R with orthogonal complement W, then: dim(W) + dim(W) n. Hint: Assemble a basis for W in the rows of a matrix and apply The Dimension Theorem for Matrices. 31. Let WR. Prove that Wn 0 Hint: Suppose i e W and i e W, What can you say about the dot product of w with itself? 32. Use the idea behind the previous Exercise to give another proof that 33. Let W g R. Our goal in this Exercise is to prove that (WW. R)0 Explain 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 W is a non-trivial subspace. Let us use the symbol U for W. Prove that W (which is W) as well as U Stare at these two definitions until you can clearly explain in writing why every vector in W also satisfies the definition of U a. cU. Hint: write down the definition of U b. c. In Exercise 39 of Section 1.7, we discussed the concept of nested subspaces. Part (b) tells t WsU SR is a nesting of subspaces. Use The Dimension Theorem for Orthogonal Complements as well as Exercise 39 of Section 1.7 to show that in fact, 31
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