Although linear representations are frequently used in image analysis, their performances are seldom optimal in specific applications. This paper proposes a stochastic gradient algorithm for finding optimal linear representations of images for use in appearance-based object recognition. Using the nearest neighbor classifier, a recognition performance function is specified and linear representations that maximize this performance are sought. For solving this optimization problem on a Grassmann manifold, a stochastic gradient algorithm utilizing intrinsic flows is introduced. Several experimental results are presented to demonstrate this algorithm.