(1) Suppose that V is a vector space and that S = {u,v} is a set...

(1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates V then T also generates V . (Hint: try solving for u and v in terms of w and x.). (c) Summarize the results of parts (a) and (b), correctly employing the word “basis”.

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Colby Messinger answered 1 year ago

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