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

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

as title

2. 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.