Subspaces of R^3...Date: 06/22/98 at 19:34:54 From: arnetta piper Subject: Subspaces of R^3... Prove that {[ a ] | a + b + c + 0 } is a subspace of R^3. [ b ] [ c ] Find a basis. Determine the dimension of the subspace. Date: 06/23/98 at 22:53:49 From: Doctor Mateo Subject: Re: Subspaces of R^3... Hello Arnetta, You have some very interesting questions here. I am going to answer them on the assumption that a + b + c + 0 is really a + b + c = 0. If this is not the case, please send the question back. In the first part of your question we are being asked to prove that _ _ | a || {| b || a + b + c = 0} (let's call this W) is a subspace of R^3. |_ c _|| We know that W is not an empty subset of R^3 since _ _ | 0 | 0 = | 0 | belongs to W since 0 + 0 + 0 = 0. |_ 0 _| ****NOTE**** Let us agree that s_1 means s(*subscript*)1 and that the underscore(_) means subscript. _ _ _ _ | s_1 | | t_1 | Now if s = | s_2 | belongs to W and t = | t_2 | belongs to W, |_ s_3 _| |_ t_3 _| then it follows that s_1 + s_2 + s_3 = 0 , t_1 + t_2 + t_3 = 0 _ _ | s_1 + t_1 | and s + t = | s_2 + t_2 | belongs to W |_ s_3 + t_3 _| since (s_1 + t_1) + (s_2 + t_2) + (s_3 + t_3) = (s_1 + s_2 + s_3) + (t_1 + t_2 + t_3) = 0 + 0 = 0. _ _ | k*s_1 | Moreover, if k is any scalar, then k*s = | k*s_2 | belongs to W |_ k*s_3 _| since k*s_1 + k*s_2 + k*s_3 = k(s_1 + s_2 + s_3) = k(0) = 0. What can you conclude since we have shown that W is closed under addition and scalar multiplication? Is W a subspace of R^3? Now let us find a basis for W and the dimension of W. _ _ | a || Since W = {| b || a + b + c = 0} |_ c _|| If we let a = p and b = q, then we have c = -p-q. Each vector v in W can then be written as _ _ _ _ _ _ | p | | 1 | | 0 | W = | q | = p| 0 | + q| 1 | |_ -p-q _| |_ -1 _| |_ -1 _| _ _ _ _ | 1 | | 0 | Therefore if we let v_1 = | 0 | and v_2 = | 1 | |_ -1 _| |_ -1 _| then W = p*v_1 + q*v_2 so that W = Span{v_1, v_2}. Thus, v_1 and v_2 are not scalar multiples of one another and therefore independent. Now that we have shown the linear independence what can you deduce about the basis of W and the corresponding dimension of W ? Hope that this helps. -Doctor Mateo, The Math Forum |
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