Let U and V be two vector spaces consisting of D-vectors over a field F.
Definition: If U and V share only the zero vector then we define the direct sum of U and V to be the set: <MATH> \{u + v : u \in U, v \in V\} </MATH> written: <MATH> U \oplus V </MATH> That is, <math>U \oplus V</math> is the set of all sums of a vector in U and a vector in V.
In Python, [u+v for u in U for v in V]
<math>dim U + dimV = dimU \oplus V</math>
Vectors over GF(2):
Let U = Span {1000, 0100} and let V = Span {0010}.
Therefore <math>U \oplus V</math> is defined.
<MATH> \begin{eqnarray*} U \oplus V & = \{& 0000+0000,& 1000+0000, &0100+0000, &1100+0000, &0000+0010, &1000+0010, &0100+0010, &1100+0010\} \\ & = \{&0000,& 1000,& 0100, &1100, &0010, &1010, &0110, &1110\} \end{eqnarray*} </MATH>
The only intersection is at the origin, so <math>U \oplus V</math> is defined.
<math>U \oplus V</math> is the set of vectors u + v where <math>u \in U \text{ and } v \in V</math>
This is just Span {[4,−1, 1], [0, 1, 1]}, Plane containing the two lines