I have a $m$ dimensional subspace $V$ of $\mathbb{R}^n$, I know for another subspace $W$ for which
$$V\oplus W=\mathbb{R}^n$$ Now basis of $V$ is given explicitly in a matrix $M$ whose $m$ collumns are linearly indipendent and play role as basis for $V$.
My question is how I can find explicitly ( when everything is given say a $n=4$, and $m=1/2/3$) a basis for $W$ from these information?
Thanks for help. If one can explain by example, it will be helpful too.
Find a basis, by the standard techniques, for the nullspace of the transpose of $M$. This will be a basis for the orthogonal complement of $V$, and the orthogonal complement of $V$ will certainly work as the $W$ you want.
Matlab suggestion: If M is a list with a basis of say $m$ column vectors for $V$, you may construct a list B of any (e.g. the canonical) $n$ basis vectors ${\Bbb R}^n$ and use A:=linalg::sumBasis(M,B) to get a basis for ${\Bbb R}^n$ again of $n$ vectors. If matlab (hopefully) calculates from left to right then the first $m$ vectors in $A$ is the given basis for $V$ (i.e. the matrix M) and the remaining $n-m$ vectors should be a basis for a complement $W$.
