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c1s4 - 1.4 Solving Systems of Linear Equations 1 Chapter 1...

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1.4 Solving Systems of Linear Equations 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.4. Solving Systems of Linear Equations Definition. A system of m linear equations in the n unknowns x 1 , x 2 , . . . , x n is a system of the form: a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = b 2 . . . . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = b m . Note. The above system can be written as Ax = b where A is the coefficient matrix and x is the vector of variables. A solution to the system is a vector s such that As = b. Defninition. The augmented matrix for the above system is [ A | b ] = a 11 a 21 · · · a 1 n b 1 a 21 a 22 · · · a 2 n b 2 . . . . . . a m 1 a m 2 · · · a mn b m .
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1.4 Solving Systems of Linear Equations 2 Note. We will perform certain operations on the augmented matrix which correspond to the following manipulations of the system of equations: 1. interchange two equations, 2. multiply an equation by a nonzero constant, 3. replace an equation by the sum of itself and a multiple of another equation. Definition. The following are elementary row operations : 1. interchange row i and row j (denoted R i R j ), 2. multiplying the i th row by a nonzero scalar s (denoted R i sR i ), and 3. adding the i th row to s times the j th row (denoted R i R i + sR j ). If matrix A can be obtained from matrix B by a series of elementary row operations, then A is row equivalent to B , denoted A B or A B . Notice. These operations correspond to the above manipulations of the equations and so:
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1.4 Solving Systems of Linear Equations 3 Theorem 1.6. Invariance of Solution Sets Under Row Equiv- alence.
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