Introduction to Linear Algebra

A comprehensive textbook providing a modern introduction to linear algebra, covering vectors, matrices, linear equations, vector spaces, orthogonality, determinants, and eigenvalues, with a focus on both theory and applications.

Master the foundations and applications of linear algebra through Gilbert Strang's intuitive and rigorous framework.

Author: Gilbert Strang

Acknowledgments: Massachusetts Institute of Technology; Wellesley-Cambridge Press

1. Compute linear combinations, dot products, vector lengths, and angles between vectors.
2. Describe the geometric configurations (lines, planes, or volumes) formed by sets of vectors.
3. Solve linear equations using matrix-vector products and interpret the role of inverse and singular matrices.
4. Differentiate between the row picture (intersecting planes) and column picture (linear combinations of vectors) of a system.
5. Execute Gaussian elimination to transform a system into an upper triangular form (U) and solve via back substitution.
6. Formalize elimination steps using elementary matrices (E_{ij}) and permutations (P_{ij}).
7. Identify whether a subset of vectors satisfies the requirements to be a subspace.
8. Perform row reduction to reach the Reduced Row Echelon Form (R) and identify the rank, pivot columns, and free variables.
9. Construct the nullspace matrix N from special solutions and describe the complete solution to linear systems.
10. Identify and prove the orthogonality of the four fundamental subspaces and determine orthogonal complements.