Ytukyg

I was studying linear algebra today, when I got a formula that gives me the dimensions of a sum between two subspaces:
$$
dim (u+w) = dim(u) + dim(w) - dim(u cap w)
$$And there's a proof below it... But before reading the proof, I wanted to give it a try... Here's what I've got:

$U$ and $W$ are subspaces.
$t = dim(U)$ and $s = dim(W)$

1)My proof for direct sum:

If i have $V = Uoplus W$, I can assume that $U cap W = 0$ and for any $z in V$ it can be written as a linear combination between the vectors of the basis $U$ and $W$:
$$
z = sum_{i=1}^{t} beta_{i}cdot U_{i} + sum_{i=1}^{s} gamma_{i}cdot W_{i} : forall U_{i} in U, W_{i} in W, beta,gamma in R
$$
Because of that two affirmatives, I can assume that the vectors in the basis $U$ and $W$ will be L.I, therefore, will be a basis for $V$, and $V$ will have: $dim(V) = dim(U) + dim(W) = t+s$.

2)My proof for sum with intersection:

Now if I have that $V = U+W$, I can assume that $U cap W neq 0 $, and because of that, there are some vectors different than the trivial one, that can be written as a linear combination of the vectors in the basis $U$ and simultaneously as a linear combination fo the vectors in the basis $W$:
$$
z = sum_{i=1}^{t} beta_{i}cdot U_{i} = sum_{i=1}^{s} gamma_{i}cdot W_{i}\ forall U_{i} in U, W_{i} in W, beta,gamma in R
$$
Now, what I think is the best to be done is to find solutions for $z$ that will give me the set of vectors that are in the intersection of $U$ and $W$.

Having in hands the numbers of vectors in the set $z$, I can see that $U+W$ will give me a L.D set (because it has some intersection $z$), compound of vectors in the basis $U$ and basis $W$, and since I know that this L.D set needs to be L.I to be a basis for V, I need to remove some dependent vectors, that are directly related to the intersection... That's why:
$$
dim (u+w) = dim(u) + dim(w) - dim(u cap w)
$$

That's what I've got by my intuition and knowledge at the moment. Please correct me because I know that I'm not being rigorous, and tell me what you think... Am I in the way? Is that a good approach to the real proof?

Thanks

To spell out what I am suggesting in my comments, for a finite dimensional vector space, $V$, with subspaces $U$ and $W$, such that $V=U+W$, but not necessarily $Ucap W = {0}$.

Let ${b_i}$ be a basis for $Ucap W$, then, in particular, since $Ucap W$ is a subspace of $U$ we have that ${b_i}$ spans a (sub)space of $U$. We construct a basis for $U$ based on the given basis for that (sub)space as follows:

If $operatorname{span}{b_i}neq U$, then we find a vector $u_1in U$ which is not in $operatorname{span}{b_i}$, and append it to the set, so we now have ${b_i,u_1}$. If this set spans $U$, we are done, other we repeat the above process to obtain a $u_2$. This process is continued until $operatorname{span}{b_i,u_j}=U$, which will happen in finitely many steps, since $U$ must be finite dimensional because $U$ is a subspace of $V$ and $V$ is finite dimensional (See note at end).

Hence the set ${b_i,u_j}$ is a basis for $U$. We perform the analogous construction for $W$, obtaining basis ${b_i,w_k}$ for $W$. Now the set ${b_i,u_j,w_k}$ must span $V$ because every vector in $V$ is a sum of a vector in $U$ with a vector in $W$ and hence can be expressed as a linear combination of the form

$v=sum_{iin I} r_ib_i+sum_{jin J}s_ju_j+sum_{kin K}t_kw_k$

(side note, all of the $s_j$'s or all of the $t_k$'s can be $0$ depending on which of $U$ or $W$ it lives in).

Now we remark that $dim V=|I|+|J|+|K|$, where $|A|$ denotes cardinality of $A$ (i.e. "size"). We further observe that $dim U=|I|+|J|$, $dim W=|I|+|K|$, and $dim (Ucap W)=|I|$. Therefore, we conclude that

$dim U+dim V-dim(Ucap W)=(|I|+|J|)+(|I|+|K|)-|I|=|I|+|J|+|K|=dim V$.

Note: I used finiteness of $V$ to construct the basis. For infinite dimensional $V$, this theorem still holds (edit: as long as $dim(Ucap V)<dim V$), if you interpret the $+$ and $-$ as cardinal arithmetic operators instead of real number arithmetic operators. But you need to use a different method to get the required basis (or a different approach to proving the theorem altogether).