linear1linear2LINEAR EQUATIONS Math 21b, O. KnillHOMEWORK 1.1: 8,14,20,24,46,26*,36* (Problems with * are recommended only and a re not turned in).SYSTEM OF LINEAR EQUATIONS. A collection of linear eq uations is called a system of linear equations.An example is3x − y − z = 0−x + 2y − z = 0−x − y + 3z = 9.It consists of three equations for three unknowns x, y, z. Linear means that no nonlinear terms likex2, x3, xy, yz3, sin(x) etc. appear. A formal definition of linear ity will be given later.LINEAR EQUATION. The equation ax+by = c is the general linear equation in two variables and ax+by +cz =d is the general linear equation in three variables. The general linear equation in n variables has the forma1x1+ a2x2+ ... + anxn= a0. Finitely many such equations fo rm a system of linear equations.SOLVING BY ELIMINATION.Eliminate variables. In the first ex ample, the first equation gives z = 3x − y. Putting this into the secondand third equation gives−x + 2y − (3x − y) = 0−x − y + 3(3x − y) = 9or−4x + 3y = 08x − 4y = 9.The first equation gives y = 4/3x and plugging this into the other equation gives 8x − 16/3x = 9 or 8x = 27which means x = 27/8. The other values y = 9/2, z = 45/8 ca n now be obtained.SOLVE BY SUITABLE SUBTRACTION.Addition of equations. If we subtract the third equation from the second, we get 3y − 4z = −9 and addthree times the second equation to the first, we get 5y − 4z = 0. Subtracting this equation to the previous onegives −2y = −9 or y = 2/9.SOLVE BY COMPUTER.Use the computer. In Mathematica:Solve[{3x − y − z == 0, −x + 2y − z == 0, −x − y + 3z == 9}, {x, y, z}] .But what did Mathematica do to solve this eq uation? We will look in this course at some efficient algorithms.GEOMETRIC SOLUTION.Each of the three equations r e presents a plane inthree-dimensional space. Points on the first planesatisfy the first e quation. The second plane is thesolution s e t to the second equation. To satisfy thefirst two equations means to be on the intersectionof these two planes which is here a line. To satisfyall three equations, we have to intersect the line withthe pla ne representing the third equation which is apoint.LINES, PLANES, HYPERPLANES.The set of points in the plane satisfying ax + by = c form a line.The set of points in space satisfying ax + by + cd = d form a plane.The set of points in n-dimensional space satisfying a1x1+ ... + anxn= a0define a set called a hyperplane.RIDDLES:”15 kids have bicycles or tricycles. Together theycount 37 wheels. How many have bicycles?”Solution. With x bicycles and y tricycles, then x +y = 15, 2x + 3y = 37. The so lution is x = 8, y = 7.”Tom, the brother of Carry has twice as many sistersas brothers while Carry has equal number of sistersand brothers . How many kids is there in total in thisfamily?”Solution If there are x brothers and y sisters, thenTom has y sisters and x −1 brothers while Carry hasx brothers and y − 1 sisters. We know y = 2(x −1), x = y − 1 so that x + 1 = 2(x − 1) and so x =3, y = 4.INTERPOLATION.Find the equation of theparabola which passes throughthe points P = (0, −1),Q = (1, 4) and R = (2, 13).Solution. Assume the parabola isgiven by the points (x, y) which satisfythe equation ax2+ bx + c = y. So, c =−1, a+b +c = 4, 4a+ 2b+c = 13. Elim-ination of c gives a +b = 5, 4a+2b = 14so that 2b = 6 and b = 3, a = 2. Theparabola has the equatio n 2x2+3x−1 =0TOMOGRAPHYHere is a toy example of a problem one has to solve for magneticresonance imaging (MRI), which makes use of the absorbtion andemission of energy in the radio frequency range of the electromag-netic s pectrum.Assume we have 4 hydrogen atoms, whose nuclei are excited withenergy intensity a, b, c, d. We measure the spin echo in 4 differentdirections. 3 = a + b,7 = c + d,5 = a + c and 5 = b + d. Whatis a, b, c, d? Solution: a = 2, b = 1, c = 3, d = 4. However,also a = 0, b = 3, c = 5, d = 2 solves the problem. This systemhas not a unique solution even so there are 4 equations and 4unknowns. A good introduction to MRI can be found online at(http://www.cis.rit.edu/htbooks/mri/inside.htm).opq ra bc dINCONSISTENT. x − y = 4, y + z = 5, x + z = 6 is a system with no solutions. It is called inconsistent.x11 x12x21 x22a11 a12 a13 a14a21 a24a31a34a41 a42 a43 a44EQUILIBRIUM. As an e xample of a system with manyvariables, consider a drum modele d by a fine net. Theheights at each interior no de needs the average theheights of the 4 neighboring nodes. The height at theboundary is fixed. With n2nodes in the interior, wehave to solve a system of n2equations. For exam-ple, for n = 2 (see left), the n2= 4 equations are4x11= a21+a12+x21+x12, 4x12= x11+x13+x22+x22,4x21= x31+x11+x22+a43, 4x22= x12+x21+a43+a34.To the right, we see the solution to a problem withn = 300, where the computer had to solve a systemwith 90′000 variables.LINEAR OR NONLINEAR?a) The ideal gas law P V = nKT for the P, V, T , the pressure p, volume V and temperature T of a gas.b) The Hook law F = k(x − a) relates the force F pulling a string extended to length x.c) Einsteins mass-energy equation E = mc2relates restmass m with the energy E of a body.ON THE HISTORY. In 2000 BC the Babylonians already studied problems which led to linear equations.In 200 BC, the Chinese used a method similar to Gaussian elimination to solve systems of linear equations.LINEAR EQUATIONS Math 21b, O. KnillHOMEWORK 1.1: 8,14,20,24,46,26*,36* (Problems with * are recommended only and a re not turned in).SYSTEM OF LINEAR EQUATIONS. A collection of linear eq uations is called a system of linear equations.An example is3x − y − z = 0−x + 2y − z = 0−x − y + 3z = 9.It consists of three equations for three unknowns x, y, z. Linear means that no nonlinear terms likex2, x3, xy, yz3, sin(x) etc. appear. A formal definition of linear ity will be given later.LINEAR EQUATION. The equation ax+by = c is the general linear equation in two variables and ax+by +cz =d is the general linear equation in three variables. The general linear equation in n variables has the forma1x1+ a2x2+ ... + anxn= a0. Finitely many such equations fo rm a system of linear equations.SOLVING BY ELIMINATION.Eliminate variables. In the first ex ample, the first equation gives z = 3x − y. Putting this into the secondand third equation gives−x + 2y −
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