- Slides: 3
Systems of Linear Equations • A system of equations of the form: a 11 x 1 + a 12 x 2 + …………. . a 1 nxn = b 1 a 21 x 1 + a 22 x 2 + …………. . a 2 nxn = b 2 : : am 1 x 1 + am 2 x 2 + …………. . amnxn = bm where the elements aij and bi are scalars and the xj are “unknown” variables is called a system of m linear equations in n unknowns. • Any (ordered) n-tuple (s 1, s 2, …. . , sn) of scalars which satisfies all of the equations is called a solution of the system. The set of all solutions is called the solution set of the system.
Matrix Formulation • A system of linear equations can be more compactly expressed in matrix notation as: Ax = b, where A = [aij] is called the coefficient matrix, and x = x 1 and b = b 1 : | : | xn bm are vectors. • Recall that a vector is an ordered k-tuple of scalars. Vectors are notated in various ways: (x 1, x 2, …. , xk) or [x 1 x 2 …. xk ] (referred to as a row vector) x 1 : xk (referred to as a column vector)
Vector Formulation • A system of linear equations can also be expressed in a vector form: x 1 v 1 + x 2 v 2 + ……. + xnvn = b, where the xi are scalar unknowns and the vi are column vectors formed from the coefficients of the original linear system. • This formulation can be interpreted as: if we can find scalars xi satisfying the equation, then the given vector b can be expressed in terms of the given vectors vi. This formulation is not useful for solving the system, but will become very important when we are working with vectors.