Ax equal to b - 1.4 The Matrix Equation Ax b Linear...

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1.4 The Matrix Equation Ax b Linear combinations can be viewed as a matrix-vector multiplication. Definition If A is an m n matrix, with columns a 1 , a 2 , , a n , and if x is in R n , then the product of A and x , denoted by A x , is the linear combination of the columns of A using the corresponding entries in x as weights . I.e., A x a 1 a 2 a n x 1 x 2 x n x 1 a 1 x 2 a 2 x n a n EXAMPLE: 1 4 3 2 0 5 7 6 7 1 3 0 6 4 2 5 7 21 0 24 12 30 31 9 30 1
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EXAMPLE: Write down the system of equations corresponding to the augmented matrix below and then express the system of equations in vector form and finally in the form A x b where b is a 3 1 vector. 2 3 4 9 3 1 0 2 Solution: Corresponding system of equations (fill-in) Vector Equation: 2 3 3 1 4 0 9 2 . Matrix equation (fill-in): 2
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Three equivalent ways of viewing a linear system: 1. as a system of linear equations; 2. as a vector equation x 1 a 1 x 2 a 2 x n a n b; or 3. as a matrix equation A x b .
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