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If you don't claim on your tax return a dependent who is an eligible student (even if entitled to claim the dependent) then only the dependent can claim the education expenses and credits on their tax return.  You can't claim the credit based on this dependent's expenses. 

As a follow-up, is the college student that can be claimed as a dependent on another's tax return able to claim the $12,000 standard deduction that they would be eligible for if another did not claim them as a dependent?

Yes, the student who is Single will have a standard deduction of $12,000 on their tax return, whether or not they were being claimed as a dependent.

TurboTax limits the standard deduction from $12,000 to $350 plus the earned income amount if earned income is less than $700, but reduces the tax due if the dependent is not claimed on another's return. I haven't been able to trace the calculation backup for the reduced tax amount.