We will talk about optimization algorithms that do not use derivatives, and discuss how to incorporate subspace techniques into this type of algorithms. We present a new derivative-free algorithm with low-dimensional subspace techniques for large-scale problems. For the new algorithm, we establish global convergence and R-linear convergence rate; we propose a preconditioning technique, which improves the performance of the algorithm on ill-conditioned problems. Our algorithm worked evidently better than NEWUOA for the problems tested, regarding the number of function evaluations and CPU time. Moreover, the new algorithm is capable of solving many 2000-dimensional test problems to high precision within several minutes, using not more than 50000 function evaluations (equivalent to less than 25 simplex gradients). This work is based on Chapter 5 of the author's PhD thesis, which was supervised by Professor Ya-xiang Yuan (Chinese Academy of Sciences, China).