Is the following reasoning that any finite dimensional vector space can be decomposed as a direct sum valid?
Since finite dimensional vector spaces are isomorphic if their dimensions match and they are defined over the same field, which I will restrict to $\Bbb R$ or $\Bbb C$, there is only one one-dimensional vector space over $\Bbb R$ and $\Bbb C$ up to isomorphism.
$\dim V=\dim U_1+\dim U_2$ if $V=U_1\oplus U_2$, therefore I can construct a vector space of any finite dimension by taking arbitrary direct sums of the unique 1-dimensional vector space over the same field.
Given now any finite dimensional vector space over $\Bbb R$ or $\Bbb C$, it will always be isomorphic as a vector space to one of the vector spaces I can construct as in step 2.
Therefore any finite dimensional vector space can be written as a direct sum of any combination of lower dimensional vector spaces provided the dimensions add up correctly.