Math 304–504Linear AlgebraLecture 2:Gaussian eliminati on.Row echelon form.System of linear equationsa11x1+ a12x2+ ··· + a1nxn= b1a21x1+ a22x2+ ··· + a2nxn= b2·········am1x1+ am2x2+ ··· + amnxn= bmHere x1, x2, . . . , xnar e variables and aij, bjar econstants.A solution of the system is a common sol uti on of allequations in the system.A system of linear equations can have one solution,infinitely many s olutions, or no solution at all.xyx − y = −22x + 3y = 6x = 0, y = 2xy2x + 3y = 22x + 3y = 6inconsistent system(no solutions)xy4x + 6y = 122x + 3y = 6⇐⇒ 2x + 3y = 6Solving systems of linear equationsElimination method always works for systems oflinear equations.Algorithm: (1) pick a variable, solve one of theequations for it, and el iminate it fro m the otherequations; (2) put aside the equation used in theelimination, and return to step (1).x − y = 2 =⇒ x = y + 22x −y − z = 5 =⇒ 2(y + 2) − y − z = 5After the elimi nati on is completed, the system issolved by back substitution.y = 1 =⇒ x = y + 2 = 3Gaussian eliminationGaussian elimination is a modi fication of theelimination method that allows only so-calledelementary operations.Elementary operations for systems of linear equations:(1) to multiply an equation by a no nzero scalar;(2) to add an equation multiplied by a s calar toanother equation;(3) to interchange two equations.Theorem Applying elementary operatio ns to asystem of linear equations does not change thesolution set of the s ystem.Operation 1: multiply the ith equation by r 6= 0.a11x1+ a12x2+ ··· + a1nxn= b1············ai1x1+ ai2x2+ ··· + ainxn= bi············am1x1+ am2x2+ ··· + amnxn= bm=⇒a11x1+ a12x2+ ··· + a1nxn= b1············(rai1)x1+ (rai2)x2+ ··· + (rain)xn= rbi············am1x1+ am2x2+ ··· + amnxn= bmTo undo the operatio n, mul tiply the ith equation by r−1.Operation 2: add r times the ith equation to thejth equation.············ai1x1+ ai2x2+ ··· + ainxn= bi············aj1x1+ aj2x2+ ··· + ajnxn= bj············=⇒············ai1x1+ ··· + ainxn= bi············(aj1+ rai1)x1+ ··· + (ajn+ rain)xn= bj+ rbi············To undo the operatio n, add −r times the ithequation to the jth equation.Operation 3: interchange the ith and jth equations.············ai1x1+ ai2x2+ ··· + ainxn= bi············aj1x1+ aj2x2+ ··· + ajnxn= bj············=⇒············aj1x1+ aj2x2+ ··· + ajnxn= bj············ai1x1+ ai2x2+ ··· + ainxn= bi············To undo the operatio n, apply it once more.Example.x − y = 22x − y − z = 3x + y + z = 6Add −2 times the 1st equation to the 2nd equation:x − y = 2y − z = −1x + y + z = 6E 2 := E 2 − 2 ∗ E 1Add −1 times the 1st equation to the 3rd equation:x − y = 2y − z = −12y + z = 4Add −2 times the 2nd equation to the 3rd equation:x − y = 2y − z = −13z = 6The elimination is completed, and we can solve thesystem by back substitution. However we may aswell proceed with elementary operations.Multiply the 3rd equation by 1/3:x − y = 2y − z = −1z = 2Add the 3rd equation to the 2nd equation:x − y = 2y = 1z = 2Add the 2nd equation to the 1st equation:x = 3y = 1z = 2System of linear equations:x − y = 22x −y − z = 3x + y + z = 6Solution: (x, y , z) = (3, 1, 2)Another example.x + y − 2z = 1y − z = 3−x + 4y − 3z = 1Add the 1st equation to the 3rd equation:x + y − 2z = 1y − z = 35y − 5z = 2Add −5 times the 2nd equation to the 3rd equation:x + y − 2z = 1y − z = 30 = −13System of linear equations:x + y −2z = 1y − z = 3−x + 4y − 3z = 1Solution: no so lution (inconsistent system).Yet another example.x + y − 2z = 1y − z = 3−x + 4y − 3z = 14Add the 1st equation to the 3rd equation:x + y − 2z = 1y − z = 35y − 5z = 15Add −5 times the 2nd equation to the 3rd equation:x + y − 2z = 1y − z = 30 = 0Add −1 times the 2nd equation to the 1st equation:x − z = −2y − z = 30 = 0⇐⇒x = z − 2y = z + 3Here z is a free variable.It follows thatx = t − 2y = t + 3z = tfor some t ∈ R.System of linear equations:x + y −2z = 1y − z = 3−x + 4y − 3z = 14Solution: (x, y , z) = (t − 2, t + 3, t), t ∈ R.In vector form, (x, y, z) = (−2, 3, 0) + t(1, 1, 1) .The set of all solutions is a line in R3passingthrough the point (−2 , 3, 0) in the direction(1, 1, 1 ).MatricesDefinition. A matrix is a rectangular ar ray of numbers .Examples:2 7−1 03 3,2 7 0.24.6 1 1,3/55/84, (√2, 0, −√3, 5),1 10 1.dimensions = (# of rows) × (# of columns)n-by-n: square matrixn-by-1: column vector1-by-n: row vectorSystem of linear equations:a11x1+ a12x2+ ··· + a1nxn= b1a21x1+ a22x2+ ··· + a2nxn= b2·········am1x1+ am2x2+ ··· + amnxn= bmCoefficient matrix and col umn vector of theright-hand sides:a11a12. . . a1na21a22. . . a2n............am1am2. . . amnb1b2...bmSystem of linear equations:a11x1+ a12x2+ ··· + a1nxn= b1a21x1+ a22x2+ ··· + a2nxn= b2·········am1x1+ am2x2+ ··· + amnxn= bmAugmented matrix:a11a12. . . a1nb1a21a22. . . a2nb2...............am1am2. . . amnbmElementary operatio ns for systems of linearequations correspond to elementary row operationsfor augmented matrices:(1) to multiply a row ( as a vector) by a nonzeroscalar;(2) to add (as a vector) the i th row multiplied (as avector) by some r ∈ R to the jth row;(3) to interchange two rows.The goal of the Gauss ian elimination is to convertthe augmented matrix into row echelon form:• all the entries below the staircase line are zero;• boxed entries, called pivotal or lead entries,ar e nonzero (variant: equal to 1);• each circled star correspond to a free variable.The original system of linear equations i s consistentif there is no leading entry in the rightmost
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