lina-06-11-09-p1 - i =1 ( v i x )( v i y ) , (b) for any x...

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Linear Algebra Geometry: Sheet 5 Set on Friday, November 6: Questions 1, 2, 3, and 4 1. Show that if the k vectors v 1 , v 2 ··· v k R n are all non-zero and mutually orthogo- nal, i.e., v i · v j = 0 if i 6 = j , then they are linearly independent. 2. Recall the definition of the Kronecker delta, δ ij := ( 0 if i 6 = j 1 if i = j , (a) Let b 1 ,b 2 ,b 3 , ··· ,b n be a set of n real numbers, show that n X i =1 b i δ ij = b j , and n X j =1 b j δ ij = b i (b) Show that n X j =1 δ ij δ jk = δ ik 3. Let V R n be a linear subspace, and v 1 , v 2 , ··· , v k an orthonormal basis of V , i.e., v i · v j = δ ij . Show that (a) for any x , y V we have x · y = k X
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Unformatted text preview: i =1 ( v i x )( v i y ) , (b) for any x V we have k x k = k X i =1 ( v i x ) 2 1 2 Apply the above formulas to the case V = R 2 and v 1 = 1 and v 2 = 1 . 4. Let V := { x R 2 ; x 1-2 x 2 = 0 } R 2 . (a) Show that V is a subspace. (b) Find a vector u V with k u k = 1. (c) Compute P V x for an arbitrary x R 2 (d) Compute the distance between x = -1 7 and V . 1...
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