Can you mine without transactions?
This is essentially correct: the protocol leaves it up to the miner to decide which transactions  to include in a block.  There is no requirement for them to include any transactions at all, other than the "coinbase" transaction which specifies where to send the block reward.

However, most transactions have fees attached, which the miner can collect only by including those transactions in the block.  So this gives the miner an incentive to include transactions if there are any outstanding.

Note that a block with no transactions does still contribute to the security of the currency: it increases the amount of hashing that an attacker would need to do to reverse a transaction recorded in a previous block.  (They would have to produce a chain showing more work than the current one, including the work proved by the zero-transaction block.)

See also Does mining activity only exist if there are bitcoin transactions?.