mathematical concepts of linear algebra in ai tools: a calculation-based study

The rapid advancement of artificial intelligence (ai) relies heavily on mathematical foundations, with linear algebra serving as a cornerstone. this paper examines the essential mathematical concepts of vector spaces, matrices, and linear transformations that underpin key ai algorithms, such as machine learning and neural networks. special attention is given to eigenvalues, eigenvectors, and matrix factorizations, including singular value decomposition (svd) and principal component analysis (pca), which are crucial for dimensionality reduction and feature extraction. additionally, the paper explores the role of quadratic programming and convex optimization in training support vector machines (svms) and deep learning models, presenting detailed mathematical formulations of these processes. computational challenges in handling large-scale matrix operations, such as multiplication, inversion, and sparse matrices, are addressed with a focus on numerical methods that enhance scalability and performance. supported by worked examples and simulations, this research bridges theoretical rigor and practical applications, offering valuable insights for advancing ai systems.