Direct Sum

$V⨁W$ is the direct sum of Vector Spaces $V$ and $W$.

It is built by constructing the space $Z:=\{v⨁w:v\in V,w\in W\}$, where $v⨁w$ can be seen as taking the Cartesian Product $V\times W$ and taking the Quotient Space with the relations spanned by:

1. $v_1⨁w_1+v_2⨁w_2=(v_1+v_2)⨁(w_1+w_2)$
2. $\lambda (v⨁w)=(\lambda v)⨁(\lambda w)$