  Reward-free reinforcement learning (RL) considers the setting where the agent
does not have access to a reward function during exploration, but must propose
a near-optimal policy for an arbitrary reward function revealed only after
exploring. In the the tabular setting, it is well known that this is a more
difficult problem than reward-aware (PAC) RL -- where the agent has access to
the reward function during exploration -- with optimal sample complexities in
the two settings differing by a factor of $|\mathcal{S}|$, the size of the
state space. We show that this separation does not exist in the setting of
linear MDPs. We first develop a computationally efficient algorithm for
reward-free RL in a $d$-dimensional linear MDP with sample complexity scaling
as $\widetilde{\mathcal{O}}(d^2 H^5/\epsilon^2)$. We then show a lower bound
with matching dimension-dependence of $\Omega(d^2 H^2/\epsilon^2)$, which holds
for the reward-aware RL setting. To our knowledge, our approach is the first
computationally efficient algorithm to achieve optimal $d$ dependence in linear
MDPs, even in the single-reward PAC setting. Our algorithm relies on a novel
procedure which efficiently traverses a linear MDP, collecting samples in any
given ``feature direction'', and enjoys a sample complexity scaling optimally
in the (linear MDP equivalent of the) maximal state visitation probability. We
show that this exploration procedure can also be applied to solve the problem
of obtaining ``well-conditioned'' covariates in linear MDPs.
