Proving undecidability of the language composed of LBAs where their language is empty
Consider the reduction that shows that LBA-emptiness is undecidable: you reduce from $\overline{A_{TM}}=\{&lt;M,w&gt;: M $ does not accept $w\}$.

So the reduction takes as input $&lt;M,w&gt;$, and outputs an LBA $T$ such that $L(T)\neq \emptyset$ iff $M$ accepts $w$.

Now, recall that an LBA is in particular a TM. So this same reduction actually outputs a TM, and actually shows that $E_{TM}$ is undecidable as well.

The reason this works is that the relationship between $E_{LBA}$ and $E_{TM}$ is such that if $M$ is an LBA, then $M\in E_{LBA}$ iff $M\in E_{TM}$. Since the reduction above always outputs an LBA, then you can just plug it in.