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Real coded genetic algorithms (RCGAs) have been widely studied and applied to deal with continuous optimization problems for years. However, how to improve the degree of accuracy so as to produce high quality solutions is still one of the main difficulties that RCGAs face with. This paper proposes a novel mutation scheme for RCGAs. The mutation operator is defined as a linear map in the space of chromosomes (in RCGAs each chromosome is a floating point vector). It operates on a whole chromosome instead of several single genes to produce the new chromosome. The linear map is represented by a randomly generated mapping matrix which satisfies some predefined constraints. By this way, the constraints restrict the mutations of genes on a same chromosome as a whole. RCGA with the proposed mutation scheme is tested on 16 benchmark functions. Results demonstrate that the proposed scheme not only improves the solution accuracy that RCGA can obtain, but also presents a very fast convergence speed. The linear map-based mutation scheme has a bright future to improve RCGAs.