Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares (free PDF)
by Stephen Boyd, Lieven Vandenberghe · Cambridge University Press / Stanford
Stanford and UCLA authors' applied linear algebra text, free PDF from Cambridge. Covers solving square and overdetermined systems via QR factorization and back substitution, with operation counts, so you understand what a solver actually does and when least squares replaces exact solution.
More resources on Systems of Linear Equations
Introduction to Linear Algebra: Systems of Linear Equations
A short lecture from Professor Dave Explains introducing linear algebra through systems of linear equations. It shows how such systems are written and solved and how they lead to matrices, giving a first footing before studying matrix methods.
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Matrix theory and linear algebra: systems of equations, elimination, vector spaces and subspaces, orthogonality and least squares, determinants, eigenvalues and positive definite matrices. Includes 35 lecture videos plus problem sets and exams with solutions, giving the fluency needed for applied mathematics, engineering and data science.
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Opening chapter of Grant Sanderson's animated linear algebra series, contrasting the physics, computer science and mathematics views of vectors. Shows vectors as arrows and coordinate lists, and how addition and scalar multiplication work geometrically, laying intuition for linear combinations and transformations.
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Fifteen-minute animated lesson recasting Ax=v as a geometric question. You will understand why a nonzero determinant guarantees a unique solution, what column space and rank say about solvability, and how the null space describes the full solution set.
Introduction to Linear Algebra, 6th Edition
The textbook behind MIT 18.06, now in its sixth edition. It builds elimination, augmented matrices, and LU factorization into the four-subspaces framework, so you can read off whether Ax=b has one solution, infinitely many, or none.