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Linear Algebra and Its Applications

Linear Algebra and Its Applications Fourth Edition Gilbert Strang x y z Ax b y Ay b b 0. z Az 0. 0. Contents Preface iv 1 Matrices and Gaussian Elimination 1. Introduction .. 1. The Geometry of Linear Equations .. 4. An Example of Gaussian Elimination .. 13. Matrix Notation and Matrix Multiplication .. 21. Triangular Factors and Row Exchanges .. 36. Inverses and Transposes .. 50. Special Matrices and Applications .. 66. Review Exercises .. 72. 2 Vector Spaces 77. Vector Spaces and Subspaces .. 77. Solving Ax = 0 and Ax = b .. 86. Linear Independence, Basis, and Dimension .. 103. The Four Fundamental Subspaces .. 115. Graphs and Networks .. 129. Linear Transformations .. 140. Review Exercises .. 154. 3 Orthogonality 159. Orthogonal Vectors and Subspaces .. 159. Cosines and Projections onto Lines .. 171. Projections and Least Squares.

Linear algebra moves steadily to n vectors in m-dimensional space. We still want combinations of the columns (in the column space). We still get m equations to produce b (one for each row). Those equations may or may not have a solution. They always have a least-squares solution. The interplay of columns and rows is the heart of linear algebra.

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