Let $W$ be a finite dimensional complex vector space and $V$ and $U$ be subspaces of $W$ with $V = \mathbb C e \oplus \mathbb C h$ and $U = \mathbb C f \oplus \mathbb C h,$ for some $e,f,h \in W.$ Then what is $V \oplus U\ $?
In my book it is written as $V \oplus U = \mathbb C e_1 \oplus \mathbb C h_1 \oplus \mathbb C e_2 \oplus \mathbb C h_2.$ But what are $e_1, e_2, h_1$ and $h_2\ $? I presume that $e_1 = (e,0,0,0), e_2 = (0,h,0,0), e_3 = (0,0,f,0)$ and $e_4 = (0,0,0,h)$ or any permutation of them would also work as the resultant external direct sums would have been isomorphic. Could anyone confirm this?
Thanks for your time.
The vectors $(e,0,0,0),(0,h,0,0),(0,0,f,0),(0,0,0,h)\in W\oplus W\oplus W\oplus W$ are irrelevant.
From the additional context (available on the page you indicate in comment), $e_1,e_2,h_1,h_2$ are respectively the copies in the external direct sum $V\oplus U$ of $e\in V,f\in U,h\in V,h\in U.$ More formally (with the notations of your document): $$e_1=i_+(e),\quad h_1=i_+(h),\quad e_2=i_-(f),\quad h_2=i_-(h).$$