Quantum computing is poised to revolutionize computational mathematics by offering exponential speedups for certain classes of problems, notably those grounded in linear algebra. Linear algebra is foundational to numerous scientific and engineering applications, including solving systems of equations, optimization, and machine learning. Quantum algorithms such as the Harrow-Hassidim-Lloyd (HHL) algorithm have demonstrated the potential for dramatic improvements over classical methods, particularly for large and sparse systems. This article surveys the landscape of quantum algorithms for linear algebra, detailing their theoretical foundations, practical implementations, and implications for future research and technology.