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Problem_Set_2 - What is the rank of T Explain 3[Basis Let S...

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T.Mitra, Fall 2008 Economics 6170 Problem Set 2 [Due on Wednesday, September 17] 1. [Linear Dependence and Independence] Let S = { e 1 , e 2 , . . . , e n } be the set of unit vectors in R n . (a) Let T = { x 1 , x 2 , . . . , x n } be a set of vectors in R n , defined by: x 1 = e 1 + e 2 , x 2 = e 2 + e 3 , . . . , x n - 1 = e n - 1 + e n , x n = e n + e 1 Is T a set of linearly independent vectors in R n ? Explain. [Hint: consider two cases, n odd, n even]. (b) Let x be an arbitrary vector in R n , with x n 6 = 0 . Let U be the set of vectors defined by U = { e 1 , e 2 , ..., e n - 1 , x } . Is U a set of linearly independent vectors in R n ? Explain. 2. [Rank] (a) Let S be a set of vectors in R n , defined by: S = { ( x 1 , ..., x n ) R n : x 1 + x 2 + · · · + x n = 2 } What is the rank of S ? Explain. (b) Let T be a set of vectors in R n , defined by: T = { ( x 1 , ..., x n ) R n : x 1 + 2 x 2 + · · ·
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Unformatted text preview: } What is the rank of T ? Explain. 3. [Basis] Let S = { x 1 ,...,x m } be a set of linearly independent vectors in R n , with m < n, and let T = { y 1 ,...,y n } be a set of basis vectors of R n . Show that there are ( n-m ) vectors in the set T, such that the m vectors in S, together with these ( n-m ) vectors in T, constitute a basis of R n . 4. [Inner Product] Suppose S = { x 1 ,...,x n } is a set of non-null vectors in R n , which are mutually orthogonal: that is, x i x j = 0 whenever i 6 = j. Show that S is linearly independent. 1...
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