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î D v } v v v U , } µ } v U dy ó ó ì ï ì ï Z µ P U Z ^ h v À Ç } ( E Á Ç U E ì ô õ ì í. U · v = v · u, (u v) · w = u · w v · w, (cu) · v = c(u · v) Further, u · u = u2 Thus, if u = 0 is the zero vector, then u · u = 0, and otherwise u · u > 0 1 Orthogonality Two vectors u and v are said to be orthogonal (perpendicular), if the angle between them is 90. Textbook solution for Multivariable Calculus 11th Edition Ron Larson Chapter 114 Problem 54E We have stepbystep solutions for your textbooks written by Bartleby experts!.

8 Let S and S0 be subsets of the vector space V such that S0 S, say S = fv 1;v 2;v 3;v 4;v 5gand S0 = fv 1;v 2;v 3g (a) If S is linearly independent, then S0 is linearly independent Assume S0 is linearly dependent Thus exists c 1;c 2;c 3 2R, not all zero, such that c 1v 1 c 2v 2 c 3v 3 = 0 v =)exists c 1;c 2;c 3;a 4;a 5 2R, not all zero, such that c 1v 1 c 2v 2 c 3v 3 a 4v 4 a 5v.

/ v r } v v v P À o o X d v Ç l ^ u Z ï ð ó r ó õ õ r í ô ó ó v Ç l X u Z v Ç X } P } } l o Ç v. Let U and V be subspaces of a vector space W We saw as a review exercise that U \V is a subspace of W, and that U V was not a subspace However, spanU V is a subspace2 Notice that all elements of spanU V take the form u v with u 2U. Ri odev kdyh d vwhzdugvkls surjudp ri wkrvh odev kdyh d surgxfwlyh dqg surjuhvvlqj frpplwwhh v p ^ µ ( µ o > } } Ç ^ Á z. Title Microsoft Word GHE Policy_ Author TTabard Created Date 3/28/17 PM.

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