In recent years, linear representation-based methods have been widely researched and applied in the image classification field. Generally speaking, there are three steps within linear representation-based classification (LRC) algorithms. The first step is coding, which uses all training samples to represent the test sample in a linear combination. The second step is subspace approximation, where residuals between the test sample and the linear combination of each class are calculated. The third step is classification, which assigns the class label to the minimum class-specific residual. We classify the LRC methods into six categories: 1) linear representation-based classification methods with norm minimizations, 2) linear representation-based classification methods with constraints, 3) linear representation-based classification methods with feature spaces, 4) linear representation-based classification methods with structural information, 5) linear representation with subspace learning, and 6) linear representation in semi-supervised learning and unsupervised learning. The purpose of this paper is to: 1) make an accurate and clear definition of the linear representation-based method, 2) provide a categorization and a comprehensive survey of the existing linear representation-based classification methods for image classification, 3) Summarize the main applications of linear representation-based methods, 4) provide extensive classification results and a discussion of the linear representation-based methods. Furthermore, this paper summarizes specific applications of the linear representation-based methods. Particularly, we performed extensive experiments to compare thirteen linear representation-based classification methods on seven image classification datasets.