To realize the high data rates with low complexity in MIMO systems, linear detections are commonly used where the received signal vector is first transformed to a new vector and then detected on per symbol basis. However, the typical linear detectors such as zero forcing detector and minimum mean-square error detector are not optimal in terms of maximizing the achievable data rate of MIMO system under linear detection constraint. In this letter, we investigate the optimal linear transformer that can maximize such rate with finite constellation inputs. By capitalizing on the relationship between mutual information and the MMSE between channel inputs and outputs, the optimal linear detector (OLD) can be expressed as a fixed-point equation and be solved iteratively. Analytical results prove the consistence between the proposed OLD and the joint maximum likelihood detector (JMLD) in the low- and high-SNR region. The further simulation results verify the consistence, and moreover, at moderate SNR, OLD can significantly improve the achievable data rate and is close to that of JMLD.