We consider the linear coding of a discrete memoryless Gaussian source transmitted through a discrete memoryless fading channel with additive white Gaussian noise (AWGN). The goal is to minimize the mean squared error (MSE) of the source reconstruction at the destination subject to an average power constraint imposed on the channel input symbols. We show that among all single-letter (or symbol-by-symbol) codes, linear coding achieves the smallest MSE, and is thus optimal. But when block length increases, the linear coding still shares the same performance with the single-letter coding, and thus can not approach the Shannon's bound. In spite of the suboptimality, the performance loss of linear coding compared to the optimal coding can be quantitively bounded in terms of the variance of the fading gain and the average transmit power. We also show that for linear coding, when there is no transmitter channel state information (CSI), uniform power allocation is optimal, and in the presence of transmitter CSI, the optimal power allocation can be analytically solved in terms of the channel fading gains and the average power budget.