Generalized Category Discovery (GCD) seeks to identify novel categories from unlabeled data while retaining the classification ability of seen categories. Prior GCD methods commonly leverage transferable representations from pre-trained models, adapting to downstream datasets via partial fine-tuning (updating only the final ViT block) and visual prompt tuning (appending learnable vectors to inputs). However, conventional partial fine-tuning offers limited flexibility, as it fails to adapt the entire model; meanwhile, visual prompt tuning is prone to overfitting, due to its sensitivity to initialization and inherently constrained capacity. To address these limitations, we propose LAGCD, a simple yet effective GCD approach that embeds a residual linear adapter into each ViT block. From the perspective of feature sparsity, we systematically show that non-linearity in conventional adapters impairs performance, whereas our linear adapter enhances it by enabling more flexible model capacity. We further introduce an auxiliary distribution alignment loss to mitigate the negative impact of biased predictions between seen and novel categories. Extensive experiments on both generic and fine-grained datasets confirm that LAGCD consistently improves performance over many sophisticated baselines. The source code is available at https://github.com/yebo0216best/LAGCD