CHAPTER I: LINEAR ALGEBRA - OCW UPM

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AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 2.SYSTEMS OF <strong>LINEAR</strong> EQUATIONS Given a system of m linear equations in the variables x 1 , x 2 , . . . , x n : ⎧ a 11 x 1 + a 12 x 2 + · · · + a 1n x n = b 1 ⎪⎨ a 21 x 1 + a 22 x 2 + · · · + a 2n x n = b 2 . ⎪⎩ a m1 x 1 + a 12 x 2 + · · · + a mn x n = b m . where m ≥ 1, the coefficients a ij and the independent terms b i , i = 1, . . . , m, j = 1, . . . , n are real numbers. The matrix equation of the system is AX = b ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a 11 a 12 · · · a 1n x 1 b 1 a A = 21 a 22 · · · a 2n ⎜ ⎟ ⎝ . . ... . ⎠ , X = x 2 ⎜ ⎟ ⎝ . ⎠ , b = b 2 ⎜ ⎟ ⎝ . ⎠ , a m1 a m2 · · · a mn x n b m where A is the the coefficient matrix, b the matrix of independent terms and X the matrix of unknowns.
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda The matrix of the system is ⎛ ⎞ a 11 a 12 · · · a 1n b 1 A | b = ⎜ a 21 a 22 · · · a 2n b 2 ⎟ ⎝ . . ... . . ⎠ . a m1 a m2 · · · a mn b m A solution of the system is a list (s 1 , s 2 , . . . , s n ) of real numbers that transform the equations in identities when substituting the values x 1 , x 2 , . . . , x n by s 1 , s 2 , . . . , s n respectively. The solution set of the system is the subset of R n given by ⎛ ⎞ s 1 {(s 1 , s 2 , . . . , s n ) ∈ R n | AS = b} where S = ⎜ s 2 ⎟ ⎝ . ⎠ . s n
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CHAPTER I: LINEAR ALGEBRA - OCW UPM

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