The direct sum of two vector spaces $~$U$~$ and $~$W,$~$ written $~$U \oplus W,$~$ is just the sum of $~$U$~$ and $~$W,$~$ but it can only be applied when $~$U$~$ and $~$W$~$ are [ linearly independent].
https://arbital.com/p/vector_space_direct_sum
by Nate Soares May 27 2016
The direct sum of two vector spaces $~$U$~$ and $~$W,$~$ written $~$U \oplus W,$~$ is just the sum of $~$U$~$ and $~$W,$~$ but it can only be applied when $~$U$~$ and $~$W$~$ are [ linearly independent].