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Vector Spaces, Basis and Closest Vector

1. Let S be a subset of R described as follows:
S {(x,y,z) :x+y+z = 0}
(a) Show that S is a vector space.
(b) Calculate a basis of S and compute it dimension.
(c) Find the vector in S which is closest to the vector (1,3, ?5) in R3.


Solution Preview

(a) For any u=(x,y,z),v=(x',y',z') in S, we have
Then u+v=(x+x',y+y',z+z') and we have
For any real number c, we have

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