subtraction of subspace?

Question

Suppose I have two subspaces $V$ and $W$ of $\mathbb{R}^n$ such that for another subspace $X$ we have $(V\cap W)\oplus X=(V+W)$

How can I find a basis explicitly for $X$ when I know basis for $V,W,V\cap W,V+W$ explicitly?

when $V,W$ is given, $V-W$ make sense?Like we know $V+W=\{x+y:x\in V, y\in W\}$

Depends upon what explicit mean? Do you have the bases in the form of matrices of colomn vectors? — H. H. Rugh, Oct 20 '16 at 19:00

score 1 · Accepted Answer · answered Oct 20 '16 at 19:17

1

If you have the bases in the form of matrices of column vectors, say $A$ for $V\cap W$ and $B$ for $V+W$ then put them together as $[A \ B]$ and make column reductions towards the right.

If $\dim(V\cap W)=k < \dim(V+W)=n$ then the first $k$ vectors is a basis of $V\cap W$ and the next (non-zero) $n-k$ vectors will give a complement (i.e. a basis for $X$) in $V+W$.

answered Oct 20 '16 at 19:17

H. H. Rugh

35,236

what does it mean by "column reductions towards the right", do you mean I need to find collumnspace([A B])? – Myshkin Oct 20 '16 at 19:20
Yes, but do you already have a basis for $V\cap W$ or do you need to find this as well? If you already have it, the easiest is probably doing Gauss elimination (but acting on columns rather than rows) on $[A\ B]$ as I suggested, to find the columnspace of $V\cap W$ and then the complement. – H. H. Rugh Oct 20 '16 at 23:23
I don't think it is that simple. Think of two planes in $R^3$ that intersects in a line. When each of the two planes are given by 2 vectors it is in general not so obvious to find a vector in the intersection? (I would probably use taking orthogonal complements but perhaps you have a better idea?) – H. H. Rugh Oct 21 '16 at 09:04
I have inbuilt commands in Matlab to find intersection of two subspaces when I know their basis – Myshkin Oct 21 '16 at 20:27
ok, that's practical. You may then simply try to compute linalg::basis(S) where $S$ is a LIST (not set) starting with the basis you found for $V\cap W$ and then all of the vectors spanning $V+W$. There are obviously too many but matlab reduces to a basis and I think (but may be wrong?) that matlab starts constructing the basis from the left, so that the one you gave for $V\cap W$ will stay and the remaining provides you the wanted complement. – H. H. Rugh Oct 21 '16 at 20:47
basis for $V+W$ is simply given by colspace(V,W), regarding them as matrix..now I need a basis for complement of $V\cap W$, by the way $V\cap W$ is sitting inside $V+W$ – Myshkin Oct 21 '16 at 20:53
yes, but my suggestion using basis(S) may actually give you a fast track way to compute the complement. – H. H. Rugh Oct 21 '16 at 20:56
I am not understanding your S, what are theere in S? – Myshkin Oct 21 '16 at 20:57
1

If V and W are two lists of column vectors, you may try to use VW:=linalg::IntBasis(V,W) to get the intersection as a list of say k vectors, and then VsumW:=linalg::sumBasis(VW,V,W) to get a basis for V+W of say n vectors. If matlab (hopefully) calculates from left to right then the first k vectors in VsumW is the basis for $V\cap W$ and the remaining $n-k$ vectors should be a basis for a complement $X$. – H. H. Rugh Oct 21 '16 at 21:03
Ah, I see!! thankx I will try tomorrow – Myshkin Oct 21 '16 at 21:05
please help http://math.stackexchange.com/questions/1977558/how-to-find-a-basis-of-complementary-subspace-of-a-subspace – Myshkin Oct 21 '16 at 21:08

subtraction of subspace?

1 Answers1