If $V_1$ and $V_2$ be two subspaces of finite-dimensional vector space $V$ such that
$\dim(V_1)=\dim(V_2)$, then there exists a subspace $W$ such that $V1⊕W=V2⊕W=V$
I really don't how to prove this result. All I know is complimentary subspaces are not necessarily unique and that if dimension of two vector spaces are equal, then they are isomorphic. But how to construct $W$ for arbitrary subspaces. Please help.
Best Answer
Let $\dim(V_1)=\dim(V_2)=m$ and $\dim(V_1\cap V_2)=r$. We can extend a basis of $V_1\cap V_2$ to a basis of $V_1$ and to a basis of $V_2$. So there are bases $v_1,\ldots,v_r,v_{r+1},\ldots,v_m$ of $V_1$ and $v_1,\ldots,v_r,v_{r+1}',\ldots,v_m'$ of $V_2$. Let $w_k=v_k+v_k'$ for $r+1\le k\le m$. Then the span $W$ of $w_{r+1},\ldots,w_m$ has $V_1+V_2=V_1\oplus W=V_2\oplus W$.
This is fine if $V_1+V_2=V$. Otherwise add a complement of $V_1+V_2$ in $V$ to $W$.