Title LA Lecture 1.pdf ✅

Linear Equations A linear equation in the variables x1, x2, . . . , xn has the form a1x1 + a2x2 + a3x3 + · · · + anxn = b where the numbers a1, . . . , an ∈ R are the equation's coecients and b ∈ R is the constant. An n-tuple (s1, s2, . . . , sn) ∈ R n is a solution of, or satises, that equation if substituting the numbers s1, . . . , sn for the variables gives a true statement: a1s1 + a2s2 + · · · + ansn = b. K.M. Naralenkov (MGIMO) Linear Algebra September 5, 2025 2 / 33
Linear Equations A system of linear equations    a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2 . . . . . . am1x1 + am2x2 + · · · + amnxn = bm has the solution (s1, s2, . . . , sn) if that n-tuple is a solution of all of the equations in the system. Note how the coecients are indexed: in the ith row the coecient of the jth variable, xj , is the number aij , and the right-hand side of the ith equation is bi . K.M. Naralenkov (MGIMO) Linear Algebra September 5, 2025 3 / 33