Low-Rank Adaptation (LoRA) is a widely adopted approach for parameter-efficient fine-tuning of large language models, enabling effective adaptation with a small number of trainable parameters. However, its reliance on linear low-rank projections restricts adaptation to linear subspaces, which can limit flexibility on complex downstream tasks. To address this, we propose RanLoRA, a Residual-aware nonlinear Low-Rank Adaptation approach that leverages the decomposition structure of pretrained weights. We used Singular Value Decomposition (SVD) to decompose pretrained weights into principal components that are kept frozen and residual components that are used for task-specific adaptation. To enhance the expressiveness of linear low-rank updates, RanLoRA incorporates a nonlinear activation layer together with a Hadamard-product-based vector modulation. This design supports an implicit progressive adaptation behavior, where optimization evolves from coarse approximation of dominant components toward residual alignment and fine-grained nonlinear refinement. Experiments on benchmarks covering commonsense reasoning, natural language understanding, image classification, and mathematical reasoning show that RanLoRA consistently outperforms vanilla LoRA and representative variants under comparable parameter budgets. These results suggest that incorporating structured nonlinearity into adapter design can enhance representational flexibility and generalization across tasks in large models.