Conventional low-rank adaptation methods build adapters without considering data context, leading to sub-optimal fine-tuning performance and severe forgetting of inherent world knowledge. In this paper, we propose context-oriented decomposition adaptation (CorDA), a novel method that initializes adapters in a task-aware manner. Concretely, we develop context-oriented singular value decomposition, where we collect covariance matrices of input activations for each linear layer using sampled data from the target task, and apply SVD to the product of weight matrix and its corresponding covariance matrix. By doing so, the task-specific capability is compacted into the principal components. Thanks to the task awareness, our method enables two optional adaptation modes, knowledge-preserved mode (KPM) and instruction-previewed mode (IPM), providing flexibility to choose between freezing the principal components to preserve their associated knowledge or adapting them to better learn a new task. We further develop CorDA++ by deriving a metric that reflects the compactness of task-specific principal components, and then introducing dynamic covariance selection and dynamic rank allocation strategies based on the same metric. The two strategies provide each layer with the most representative covariance matrix and a proper rank allocation. Experimental results show that CorDA++ outperforms CorDA by a significant margin. CorDA++ in KPM not only achieves better fine-tuning performance than LoRA, but also mitigates the forgetting of pre-trained knowledge in both large language models and vision language models. For IPM, our method exhibits faster convergence, \emph{e.g.,} 4.5x speedup over QLoRA, and improves adaptation performance in various scenarios, outperforming strong baseline methods. Our method has been integrated into the PEFT library developed by Hugging Face.