Intuition behind elementary result in Linear Algebra What is the intuition behind the following result in linear Algebra

Given that $V$ is finite-dimensional and $U$ is a subspace of $V$. Then there is a subspace $W$ in $V$ such that $V=U\oplus W$.

Kindly in your answers do not mention the proof. I am aware of the proof, I merely wish to know what motivates the above result. Examples would be appreciated.
 A: If you have a subspace $U$ of a finite dimensional vector space $V$, say $\dim V = n$, $U$ has finite dimension $m\le n$. Fix a basis of $U$. We may extend this basis to a basis of $V$. $V$ and $U$, then, "differ" by $n-m$ basis vectors. $W$ is the subspace generated by the basis vectors you lose in moving from $V$ to $U$.
I hope this is sufficiently non-awkward and helpful.
A: Here is a basic illustration. Let $V = \mathbb{R}^3$ and let $U$ be any line in $V$ passing through the origin. This means you can find another subspace (a plane) perpendicular to your line, which will characterize the entire space.
Similarly if you pick a plane, there will be a line perpendicular to it to characterize the space.
A: Think of vector spaces as sets of arrows pointing where you can go.

In my drawing, you have a point O from which you start. Now E is the whole 3 dimensional space, while, 3 in this drawing is just the flat plane. If you can only go along the vectors that are in 3, you won't be able to leave the plane, so you need to add another direction, which will be a line U. 
We denote this as $E=U+W$. This isn't however the only thing we need: we need this sum to be direct. Direct sum essentially mean, that the choice of vectors in U and W is unique for each point you want to get to.
Now if you want to get to let's say a point A, you can first go along the vectors in U and then you just go along a direction parallel to the line U, which correspond to vectors in U. If we took U to be just another plane, we would get an infinity of choices how to get to the point A. I have illustrated two of them: 

The point of this theorem is therefore that if you choose a subspace W, then you can always find a space W, such that you can get everywhere in a unique way, combining vectors of W and U. U is then called a supplement of W. In case W is a hyperplane, U will for example be a line.
I hope this sheds some light on this rather important result.
