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6.2 #3 optional
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- Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. a Explain what it means to say that a set of vectors is linearly independent. b Determine whether the set S is linearly dependent or independent. S={(1,0,1,0),(0,3,0,1),(1,1,2,2),(3,4,1,2)}Guided Proof Prove that if w is orthogonal to each vector in S={v1,v2,,vn}, then w is orthogonal to every linear combination of vector in S. Getting Started: To prove that w is orthogonal to every linear combination of vectors in S, you need to show that their inner product is 0. i Write v as a linear combination of vectors, with arbitrary scalars c1,,cn in S. ii Form the inner product of w and v. iii Use the properties of inner products to rewrite the inner product w,v as a linear combination of the inner products w,vi, i=1,,n. iv Use the fact that w is orthogonal to each vector in S to lead to the conclusion that w is orthogonal to v.Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.
- Proof Let {v1,v2,...,vn} be a linearly independent set of vectors in a vector space V. Delete the vector vk from this set and prove that the set {v1,v2,...,vk1} cannot span V.Proof When the set of vectors {u1,u2,...,un} is linearly independent and the set {u1,u2,...,un,v} is linearly dependent, prove that v is the linear combination of the uis.True or False? In Exercises 79 and 80, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows that the statement is not true in all cases or cite an appropriate statement from the text. a If dim(V)=n, then any set of n+1 vectors in V must be linearly dependent. b If dim(V)=n, then any set of n1 vectors in V must be linearly independent.
- Proof Prove that if S={v1,v2,,vn} is a basis for a vector space V and c is a nonzero scalar, then the set S1={cv1,cv2,,cvn} is also a basis for V.True or false? In Exercises 43and 44, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The orthogonal complement of Rn is empty set. (b) If each vector vRn can be uniquely written as a sum of a vector s1 from S1 and a vector s2 from S2, then Rn is direct sum of S1 and S2.