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Explore fundamental concepts of linear algebra through this comprehensive 9-hour video series covering systems of equations, matrix operations, vector spaces, and advanced topics. Begin with systems of equations and matrix notation, then progress through solving linear systems using echelon form and pivots. Master vector operations including addition, multiplication, linear combinations, and span while learning to work with matrix-vector equations and the identity matrix. Delve into homogeneous and nonhomogeneous systems, linear independence, and transformations including geometric, injective, and surjective types. Study matrix operations in depth including addition, scaling, multiplication, transposes, and inverse matrices with practical solving techniques. Advance to vector spaces and subspaces with rigorous proof examples, exploring null spaces, column spaces, kernels, and ranges of linear transformations. Understand linear independence, bases, spanning sets, and coordinate systems including change of basis transformations. Examine dimension theory, row spaces, and the rank theorem with extensive proof examples and computational practice. Conclude with determinant calculations using cofactor expansion, determinant properties, and eigenvalue problems including eigenvectors, eigenspaces, characteristic equations, and matrix diagonalization techniques.
