Linear algebra is the branch of mathematics concerned with vector spaces, linear transformations, and systems of linear equations. It provides the foundational language for countless fields—from solving differential equations and optimizing machine learning models to rendering computer graphics and analyzing networks. At its core, linear algebra studies how linear combinations of vectors span spaces, how matrices encode transformations, and how eigenvalues reveal fundamental structures. Understanding that matrices represent linear maps between vector spaces—not just grids of numbers—unlocks the deeper insights that make this subject so powerful across science and engineering.

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