Linear algebra Learning Note three: Elementary transformation of matrices and linear equations

Reference: "Linear algebra" Tongji University, fourth edition

1. Elementary transformations of matrices

1) definition

2) equivalence relationship between matrices: definition, Nature (3)

3) row ladder-shaped matrix; the simplest form; standard type; equivalence class

2. Elementary matrices

1) definition

2) Theorem 1 (Relationship between elementary transformation and elementary matrix)

3) theorem 2 (the necessary and sufficient conditions for the reversible matrix A) and its inference (2, the necessary and sufficient conditions for the reversible square matrices, the necessary and sufficient conditions for the equivalence of M*n matrixes A and B)

4) Solving equations

3. Rank of matrix

1) Definition: K-order subtype; rank of matrix; full rank matrix; descending rank matrix

2) Theorem 3: The relationship between the rank of equivalent matrices

3) The nature of the rank of the matrix (8 articles)

4. Solution of linear Equation Group

1) compatible and incompatible linear systems

2) Theorem 4

3) solving systems of linear equations

4) Theorem 5: The necessary and sufficient conditions for the solution of ax=b of linear systems

5) theorem 6:n linear equation Group ax = 0 A sufficient and necessary condition for non-0 solutions

6) Theorem 7 (theorem 5 generalization): the necessary and sufficient conditions for the solution of ax=b matrix equation

7) theorem 8:ab=c, R (C) <= min{r (A), R (B)}

8) Theorem 9 (theorem 6 generalization):