hkus10 Posted March 21, 2011 Share Posted March 21, 2011 (edited) 1) Let u and v be nonzero vectors in a vector space V. show that u and v are linearly dependent if and only if there is a scalar k such that v = ku. Equivalently, u and v are linearly independent if and only if neither vector is a multiple of the other. 2) Let S = {v1, v2, ..., vk} be a set of vectors in a vector space V. Prove that S is linearly dependent if and only if one of the vectors in S is a linear combination of all the other vectors in S. For these two questions, I know I have to prove them in both directions because of "if and only of". However, how to approach this problem? what Thms or definition should I use to prove them? 3) Let S = {v1, v2, ..., vk} be a set of vectors in a vector space V, and let W be a subspace of V containing S. Show that W contains span S. For question 3, does "W be a subspace of V containing S" mean W contains S? If yes, what is the reason to show it? Edited March 21, 2011 by hkus10 Link to comment Share on other sites More sharing options...
DrRocket Posted March 21, 2011 Share Posted March 21, 2011 1) Let u and v be nonzero vectors in a vector space V. show that u and v are linearly dependent if and only if there is a scalar k such that v = ku. Equivalently, u and v are linearly independent if and only if neither vector is a multiple of the other. 2) Let S = {v1, v2, ..., vk} be a set of vectors in a vector space V. Prove that S is linearly dependent if and only if one of the vectors in S is a linear combination of all the other vectors in S. For these two questions, I know I have to prove them in both directions because of "if and only of". However, how to approach this problem? what Thms or definition should I use to prove them? 3) Let S = {v1, v2, ..., vk} be a set of vectors in a vector space V, and let W be a subspace of V containing S. Show that W contains span S. For question 3, does "W be a subspace of V containing S" mean W contains S? If yes, what is the reason to show it? For 1 & 2 all that you need is the definition of linear independence. For 3 you need the definitions of "span" and of "vector space". The statement "let W be a subspace of V containing S' means W contains S and W is a subspace of V. Link to comment Share on other sites More sharing options...
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