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  3. problem 17 and 18...

Question: problem 17 and 18...

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Theorem 1.6.17 Let β = v1,V2. . . . ,Vr be a list of vectors in a nonzero F-vector space (a) Suppose that β is linearly independent and does not span V v e V and v span β. then the list V...Vr. V is linearly independent. Suppose that β is linearly dependent and span β :: V. İf cm + c2v2 +--+ crv,- a nontrivial linear combination and j e (1.2,...r) is any index such that cj 0, then Vi,v2V...v spans V 0 is (b) 0, des a Cvi e civi E Proof (a) Suppose that civi+cv2+.tov,tev -0.Ifc 0, then nv2 v.), which is a contradiction. Thus, c 0, and hence civi+22 =0. The linear independence of β implies that C1 = c2 = . .-cr = 0. We that the list vI, 2. . V,, v is linearly independent. (b) If r 1, then the linear dependence of the list v implies that v 0, so v span β-(0}, which is a contradiction. Thus, r > 2. Since ci 0, we have vi =-ci ctor 12) Ni This identity can be used to eliminate v, from any linear combination in appears, so any vector that is a linear combination of the r vectors in the list B (namely, every vector in V) is also a linear combination of the r-I vectors in the list vi,v...V 13) 1.7 Problems ection 1 .2, explain how V-C can be thought of as . Define v + w = Show that )v with these two operations is a real vector space. What is the zero vector P1.3 Show that the intersection of any (possibly infinite) collection of subspaces of an In the spirit of the examples in S a vector space over IR. Is P.1.1 R a vector space over C? P.1.2 Let V be the set of real 2 x 2 matrices of the form v 1 cv (ordinary matrix multiplication) and cv - st it in V? F-vector space is a subspace. P.1.4 Let u be a subset of an F-vector space V. Show that spanU is the intersection of all P.1.5 Let u and W be subspaces of an F-vector space V. Prove that u UW is a subspace P.1.6 P.1.7 ? the subspaces of V that contain I. What does this say ifu of V if and only if either u S W or W S u Give an example of a linearly dependent list of three vectors in F3 such that any two of them are linearly independent. Let n > 2 be a positive integer and let A e Mn(C). Which of the following subsets of Mn (C) is a subspace of the complex vector space Mn(C)? Why? (a) All invertible matrices; (b) All noninvertible matrices; (c) All A such that A2 0; (d) All matrices whose first column are zero; (e) All lower triangular matrices; (f) All X M,(C) such that AX + XTA = 0. Let V be a real vector space and suppose that the list β linearly independent. Show that the list y uV, v-w, w+u is linearly independent. What about the list 8u + v, v+ w, w +u? u, v, w of vectors in V is P. 1.8

problem 1.7 and 1.8

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