wlog one can assume that $||E_1||=1$.

Let $X$ be the span of the matrix units $E_{11},E_{21},E_{31}$. Then for every $2$ linearly independent operators $E_1, E_2$ in this space there exists a unitary matrix $U$ such that $UE_1=E_{11}$, $UE_2=\lambda E_{21}+E_{11}$. But $E_{11}$ and $E_{12}$ are not simultaneously singular valued decomposable.

wlog one can assume that $||E_1||=1$.

Let $X$ be the span of the matrix units $E_{11},E_{21},E_{31}$. Then for every $2$ linearly independent operators $E_1, E_2$ in this space there exists a unitary matrix $U$ such that $UE_1=E_{11}$, $UE_2=\lambda E_{21}+E_{11}$. But $E_{11}$ and $\lambda E_{21}+E_{11}$ are not simultaneously singular valued decomposable.

wlog one can assume that $||E_1||=||E_2||=1$.

Let $X$ be the span of the matrix units $E_{11},E_{21},E_{31}$. Then for every $2$ linearly independent operators $E_1, E_2$ in this space there exists a unitary matrix $U$ such that $UE_1=E_{11}$, $UE_2=E_{21}$. But $E_{11}$ and $E_{12}$ are not simultaneously singular valued decomposable.

wlog one can assume that $||E_1||=1$.

Let $X$ be the span of the matrix units $E_{11},E_{21},E_{31}$. Then for every $2$ linearly independent operators $E_1, E_2$ in this space there exists a unitary matrix $U$ such that $UE_1=E_{11}$, $UE_2=\lambda E_{21}+E_{11}$. But $E_{11}$ and $E_{12}$ are not simultaneously singular valued decomposable.