[理学]线性代数II-chapter21-3.ppt

Example 6 Solution 3. The relationship between nonsingular matrix and elementary matrix Theorem 5 Proof The sufficient condition is obviously. Theorem 6 Proof Example 7 Solution * Prof Liubiyu New words elementary operation, 初等变换 elementary row operation 初等行变换 elementary column operation 初等列变换 elementary matrix 初等矩阵 row echelon form 行阶梯形 standard form 标准形 Contents §1.1-1.3 Elementary operations of matrices, normal forms and ranks of matrices §1.4 n-dimensional vectors and linear dependence §1.5 n-dimensional vector space Purpose of teaching To mastery the elementary operation of a matrix To know the properties of the elementary matrix To know the concept of equivalence between matrices (4) To understand the concept of the rank of a matrix (5) To master how to find the rank of a matrix §1.1-1.3 Elementary operations of matrices, normal forms and ranks of matrices Definition 1 (Elementary row operations) There are three types of elementary row operations that can be performed on matrices (1) Interchanges two rows (ri?rj interchanges row i and j) (2) Multiplying a row by a nonzero scalar (ri?kri multiplies row i by the nonzero scalar k) (3) Adding a multiple of one row to another row (rj? rj +kri adds k times row i to row j) 1. The elementary operation of matrix Notaton1: Similarly, we can define the elementary column operations Notation 2: The elementary both row and column operations are called the elementary operations Theorem 1 Notation 3: We now describe the procedure to transform a given matrix into row echelon form or reduced row echelon form by performing suitable elementary row operations. Step 1: Select one of the rows that has its first entry nonzero. If this happens to be the first row, then nothing needs to be done. Otherwise, interchange the selected row, say the ith, and the first row. Step 4: Ignoring temporarily the first row and first column, repeat the above process on the matrix obtained by disregarding the first row and