u1 - Unit I: Systems of Linear Equations 1. Homogeneous...

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Unit I: Systems of Linear Equations 1. Homogeneous Systems vectors, the vector space R n , addition of vectors, multiplication of vectors by scalars, linear combinations of vectors homogeneous system of linear equations, coefficient matrix of the system Gauss-Jordan elimination algorithm, pivot, basic (leading) variable, free variable, form of general solution Theorem 1. If a homogeneous system has more unknowns than equations, then there is a nontrivial (nonzero) solution of the system. Gaussian elimination with backsolving The Gauss-Jordan elimination algorithm provides a clear idea of the structure of the so- lution set for the system. However for actually solving the system, it is usually more efficient to use Gaussian elimination with backsolving, as it requires fewer arithmetic operations. 2. Matrix Notation Let A be an m × n matrix. We write A = [ a 1 ,..., a n ], where a 1 ,..., a n are the column vectors of A , and we define A x = a 1 x 1 + a 2 x 2 + ··· + a n x n . Thus A x is a linear combination of the columns of A . With this notation, the matrix form of the linear homogeneous system of equations corresponding to A is simply A x = 0 . The matrix A operates linearly on vectors: A ( x + y ) = A x + A y and A ( c x ) = cA ( x ). Linearity is equivalent to:
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u1 - Unit I: Systems of Linear Equations 1. Homogeneous...

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