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Unformatted text preview: Two vectors in R n are said to be parallel or collinear if one of them is a scalar multiple of the other. Example: 1. 0 is parallel to every vector in R n 2. (2,6) is parallel to (1,3) Forming new vectors from old ones: Addition, substraction and scalar multiplication are used in combination to form new vectors. Example: Let u, w R 3 . Then u + w R 3 ,u- 3 w R 3 . Hence we obtained two new vectors from u and w. Definition: A vector w R n is said to be linear combination of the vectors v 1 ,v 2 ,...,v k in R n if w can be expressed in the form w = c 1 v 1 + c 2 v 2 + ... + c k v k The scalars c 1 ,c 2 ,...,c k are called coefficients in the linear combination. Example: RGB color model r = (1 , , 0) pure red, g = (0 , 1 , 0) (pure green), b = (0 , , 1) (pure blue) c 1 r + c 2 g + c 3 b gives different colors, c i = percentage of the corresponding color Example: Let u = (- 3 , 1 , 2) ,v = (4 , ,- 8) ,w = (6 ,- 1 ,- 4) . Find the compo- nents of the vector x that satisfies 2u-v+x=7x+w...
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- Spring '11