Chapter 1.1 SystemsLinearEquations

0 1 0 0 1 3 1 0 f 04 0 2 11 011 100 1203 g 0

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Unformatted text preview: 13 00 1 2 00 in row echelon form? 0 1 0 0 1 3 1 0 F= 04 0 2 11 011 100 1203 G = 0 0 1 0 0000 Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Gaussian Elimination The elementary row operations that can be performed on a matrix A are: 1 2 3 Interchange two rows of A. Replace a row r of A with λr , where λ is a nonzero real number. Replace a row r of A with r + λr , where λ ∈ R and r is another row of A. Gaussian elimination is the process of using elementary row operations to change a matrix A into a matrix R , with R in row echelon form. One can solve a system of linear equations using Gaussian elimination together with back substitution. Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Examples Example Use Gaussian elimination and back substitution to solve the linear systems. 1 3x1 + 2x2 − x3 = 4 x1 − 2x2 + 2x3 = 1 11x1 + 2x2 + 4x3 = 14 2 x1 + x2 = 1 x1 − x2 = 3 −x1 + 2x2 = −2 (This system is overdetermined) Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Reduced Row Echelon Form of a Matrix A matrix A is said to be in reduced row-echelon form if A has the following properties. 1 2 A is in row echelon form. If row i is nonzero with leading 1 in column j , then column j has only one nonzero entry, namely the leading 1 in row i . Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Examples Which of 1 0 A= 0 2 0 B= 0 0 C = 0 0 the following matrices are 23 1 2 12 01 0 0 D= 0 0 03 00 1 2 00 10 0 1 E= 13 00 1 2 00 in reduced row echelon form? 0 1 0 0 1 3 1 0 F= 04 0 2 11 011 100 1203 G = 0 0 1 0 0000 Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Gauss-Jordan Elimination Gauss-Jordan elimination is the process of using elementary row operations to change a matrix A into a matrix R , with R in reduced row echelon form. The reduced row ech...
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