I have $V,W$ subspace of $\mathbb{R}^n$ of dimension $n_1,n_2$, we know $V\cap W\subseteq V$ as well as $\subseteq W$ $\Rightarrow V\cap W\subseteq V+W$, now suppose I have a $n\times n_1$ matrix M whose columns are linearly indipendent and represents a basis for $V$, suppose I have a $n\times n_2$ matrix N whose columns are linearly indipendent and represents a basis for $W$, I have a basis for $V\cap W$ with the help of $M$ and $N$ say represented by linearly indipendent columns of matrix $J=int(V\cap W)$, I also have a basis for $V+W$ represented by lin indip columns of the matrix $Q$, now my question is
since $V\cap W\oplus R_1=V+W$, for some $R_1$, I want this $R_1$ in the matrix form.
$\dim(V\cap W)<\dim (V+W)$ so there must be some linearly indipendent vectors which I add to a basis of $V\cap W$ will give me a basis for $V+W$, I want those few vectos in a matrix form.