Modern natural language systems rely on large language models, whose sheer size makes full fine-tuning costly in computation, graphics processing unit (GPU) memory, and storage. Low-rank adaptation (LoRA) sidesteps most of that cost. It keeps the pre-trained weights frozen and captures each task-specific change as the product of two smaller matrices, so adapting a model reduces to a low-rank decomposition. This review covers LoRA and its main variants and pays particular attention to the linear algebra behind them. It first explains why the low intrinsic dimension of fine-tuning makes low-rank updates effective, then compares the major variants: quantized LoRA (QLoRA), quantization-aware LoRA (QA-LoRA), adaptive low-rank adaptation (AdaLoRA), sparse low-rank adaptation (SoRA), and weight-decomposed low-rank adaptation (DoRA). Across published studies, these methods come close to full fine-tuning accuracy while updating well under one percent of a model's parameters in some settings. For reference, LoRA cuts the trainable parameter count of Generative Pre-trained Transformer 3 (GPT-3) by four orders of magnitude, and QLoRA brings a 65-billion-parameter model within the memory of one 48 GB card. Open problems remain in choosing the rank, comparing results across studies, limiting quantization loss, and combining multiple adapters without interference. Ultimately, an established piece of linear algebra, approximating high-dimensional objects in low-dimensional subspaces, is what keeps the adaptation of very large models affordable.