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Math Help - How to prove two subspaces with same dimension will have a common complement

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    How to prove two subspaces with same dimension will have a common complement

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  2. #2
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    Here's a sketch of a proof.

    Let U and V be the subspaces. If U≠V then neither of U and V can contain the other. So there exist vectors u\in U\setminus V and v\in V\setminus U. Then w_1\mathrel{\mathop=^{\mathrm{d{}ef}}}u+v is in neither U nor V. If U=V then take w_1 to be any vector not in U.

    Now proceed by an inductive construction. Let U_1 be the subspace spanned by U and w_1, and let V_1 be the subspace spanned by V and w_1. These spaces have the same dimension as each other. If they are not the whole space then we can repeat the construction in the previous paragraph to get a vector w_2 that is in neither U_1 nor V_1. Continue doing this, getting a sequence of vectors w_1, w_2, w_3,\ldots. On the assumption that the ambient spaces is finite-dimensional, this construction will stop after a finite number of steps. The subspace spanned by w_1, w_2, w_3,\ldots will then be complementary for both U and V.
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