Let V be a vector space. Let U and W be subspaces of V. Let the intersection of U and W be limited to the zero vector.
V is a direct sum of U and W if:
Every vector v in V can be uniquely expressed as v = u + w.
The word ‘can’ might be misleading. Everything is such that for each possible v, we can find only one combination of u and w that will give us the equality below:
u + w = v
For our example:
Let the vector space U be a number line with the coordinate 
All vectors from U will be of the form 
Let the vector space W be a plane with two coordinates 
All vectors from W will be of the form 
Let v=(2,5,7)
We can only get this with
(2,0,0) + (0,5,7) = (2,5,7)
Vectoreverything 