LINEAR ALGEBRA
C H A P T E R SYSTEMS OF LINEAR EQUATIONS AND MATRICES CHAPTER OUTLINE SYSTEMS OF LINEAR EQUATIONS MATRICES AND ELEMENTARY ROW OPERATIONS THE INVERSE OF A SQUARE MATRIX ELEMENTARY MATRICES AND LU FACTORIZATION APPLICATIONS OF SYSTEMS OF LINEAR EQUATIONS N THE PROCESS OF PHOTOSYNTHESIS SOLAR ENERGY IS CONVERTED INTO FORMS THAT ARE USED BY LIVING ORGANISMS THE CHEMICAL REACTION THAT OCCURS IN THE LEAVES OF PLANTS CONVERTS CARBON DIOXIDE AND WATER TO CARBOHYDRATES WITH THE RELEASE OF OXYGEN THE CHEMICAL EQUATION OF THE REACTION TAKES THE ACO2 BH2O CO2 DC6H12O6 WHERE A B C AND D ARE SOME POSITIVE WHOLE NUMBERS THE LAW OF CONSERVATION OF MASS STATES THAT THE TOTAL MASS OF ALL SUBSTANCES PRESENT BEFORE AND AFTER A CHEMICAL REACTION REMAINS THE SAME THAT IS ATOMS ARE NEITHER CREATED NOR DESTROYED PHOTOGRAPH BY JAN SMITH RF IN A CHEMICAL REACTION SO CHEMICAL EQUATIONS MUST BE BALANCED TO BALANCE THE PHO TOSYNTHESIS REACTION EQUATION THE SAME NUMBER OF CARBON ATOMS MUST APPEAR ON BOTH SIDES OF THE EQUATION SO THE SAME NUMBER OF OXYGEN ATOMS MUST APPEAR ON BOTH SIDES SO 2A B 2C 6D AND THE SAME NUMBER OF HYDROGEN ATOMS MUST APPEAR ON BOTH SIDES SO THIS GIVES US THE SYSTEM OF THREE LINEAR EQUATIONS IN FOUR VARIABLES ANY POSITIVE INTEGERS A B C AND D THAT SATISFY ALL THREE EQUATIONS ARE A SOLUTION TO THIS SYSTEM WHICH BALANCES THE CHEMICAL EQUATION FOR EXAMPLE A 6 B 6 C AND D 1 BALANCES THE EQUATION MANY DIVERSE APPLICATIONS ARE MODELED BY SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS ARE ALSO IMPORTANT IN MATHEMATICS AND IN PARTICULAR IN LINEAR ALGEBRA IN THIS CHAPTER WE DEVELOP SYSTEMATIC METHODS FOR SOLVING SYSTEMS OF LINEAR EQUATIONS AS THE INTRODUCTORY EXAMPLE ILLUSTRATES MANY NATURALLY OCCURRING PROCESSES ARE MODELED USING MORE THAN ONE EQUATION AND CAN REQUIRE MANY EQUATIONS IN MANY VARI ABLES FOR ANOTHER EXAMPLE MODELS OF THE ECONOMY CONTAIN THOUSANDS OF EQUATIONS AND THOUSANDS OF VARIABLES TO DEVELOP THIS IDEA CONSIDER THE SET OF EQUATIONS 2X Y X 2Y WHICH IS A SYSTEM OF TWO EQUATIONS IN THE COMMON VARIABLES X AND Y A SOLUTION TO THIS SYSTEM CONSISTS OF VALUES FOR X AND Y THAT SIMULTANEOUSLY SATISFY EACH EQUATION IN THIS EXAMPLE WE PROCEED BY SOLVING THE RST EQUATION FOR Y SO THAT Y 2X TO ND THE SOLUTION SUBSTITUTE Y 2X 2 INTO THE SECOND EQUATION TO OBTAIN X 2 2X 2 AND SOLVING FOR X GIVES SUBSTITUTING X 2 BACK INTO THE RST EQUATION YIELDS 2 2 Y 2 SO THAT Y THEREFORE THE UNIQUE SOLUTION TO THE SYSTEM IS X 2 Y 2 SINCE BOTH OF THESE EQUATIONS REPRESENT STRAIGHT LINES A SOLUTION EXISTS PROVIDED THAT THE LINES INTERSECT THESE LINES INTERSECT AT THE UNIQUE POINT 2 2 AS SHOWN IN FIG 1 A A SYSTEM OF EQUATIONS IS CONSISTENT IF THERE IS AT LEAST ONE SOLUTION TO THE SYSTEM IF THERE ARE NO SOLUTIONS THE SYSTEM IS INCONSISTENT IN THE CASE OF SYSTEMS OF TWO LINEAR EQUATIONS WITH TWO VARIABLES THERE ARE THREE POSSIBILITIES 1 THE TWO LINES HAVE DIFFERENT SLOPES AND HENCE INTERSECT AT A UNIQUE POINT AS SHOWN IN FIG 1 A 2 THE TWO LINES ARE IDENTICAL ONE EQUATION IS A NONZERO MULTIPLE OF THE OTHER SO THERE ARE IN NITELY MANY SOLUTIONS AS SHOWN IN FIG 1 B X Y 3 THE TWO LINES ARE PARALLEL HAVE THE SAME SLOPE AND DO NOT INTERSECT SO THE SYSTEM IS INCONSISTENT AS SHOWN IN FIG 1 C WHEN WE ARE DEALING WITH MANY VARIABLES THE STANDARD METHOD OF REPRESENTING LINEAR EQUATIONS IS TO AF X SUBSCRIPTS TO COEF CIENTS AND VARIABLES A LINEAR EQUATION IN THE N VARIABLES X1 X2 XN IS AN EQUATION OF THE FORM A1X1 A2X2 ANXN B TO REPRESENT A SYSTEM OF M LINEAR EQUATIONS IN N VARIABLES TWO SUBSCRIPTS ARE USED FOR EACH COEF CIENT THE RST SUBSCRIPT INDICATES THE EQUATION NUMBER WHILE THE SECOND SPECI ES THE TERM OF THE EQUATION SYSTEM OF LINEAR EQUATIONS A SYSTEM OF M LINEAR EQUATIONS IN N VARIABLES OR A LINEAR SYSTEM IS A COLLECTION OF EQUATIONS OF THE FORM A11X1 A12X2 A1NXN B1 A21X1 A22X2 A2NXN B2 A31X1 A32X2 A3NXN B3 AM1X1 AM2X2 AMNXN BM THIS IS ALSO REFERRED TO AS AN M N LINEAR SYSTEM FOR EXAMPLE THE COLLECTION OF EQUATIONS IS A LINEAR SYSTEM OF THREE EQUATIONS IN FOUR VARIABLES OR A 3 4 LINEAR SYSTEM A SOLUTION TO A LINEAR SYSTEM WITH N VARIABLES IS AN ORDERED SEQUENCE S1 S2 SN SUCH THAT EACH EQUATION IS SATIS ED FOR X1 S1 X2 S2 XN SN THE GENERAL SOLUTION OR SOLUTION SET IS THE SET OF ALL POSSIBLE SOLUTIONS THE ELIMINATION METHOD THE ELIMINATION METHOD ALSO CALLED GAUSSIAN ELIMINATION IS AN ALGORITHM USED TO SOLVE LINEAR SYSTEMS TO DESCRIBE THIS ALGORITHM WE RST INTRODUCE THE TRIANGULAR FORM OF A LINEAR SYSTEM AN M N LINEAR SYSTEM IS IN TRIANGULAR FORM PROVIDED THAT THE COEF CIENTS AIJ 0 WHENEVER I J IN THIS CASE WE REFER TO THE LINEAR SYSTEM AS A TRIANGULAR SYSTEM TWO EXAMPLES OF TRIANGULAR SYSTEMS ARE X1 X2 X3 X4 WHEN A LINEAR SYSTEM IS IN TRIANGULAR FORM THEN THE SOLUTION SET CAN BE OBTAINED USING A TECHNIQUE CALLED BACK SUBSTITUTION TO ILLUSTRATE THIS TECHNIQUE CONSIDER THE LINEAR SYSTEM GIVEN BY FROM THE LAST EQUATION WE SEE THAT X3 2 SUBSTITUTING THIS INTO THE SECOND EQUATION WE OBTAIN X2 3 2 5 SO X2 11 FINALLY USING THESE VALUES IN THE RST EQUATION WE HAVE X1 2 11 2 1 SO X1 19 THE SOLUTION IS ALSO WRITTEN AS 19 11 EQUIVALENT LINEAR SYSTEMS TWO LINEAR SYSTEMS ARE EQUIVALENT IF THEY HAVE THE SAME SOLUTIONS FOR EXAMPLE THE SYSTEM HAS THE UNIQUE SOLUTION X1 19 X2 11 AND X3 2 SO THE LINEAR SYSTEMS THE NEXT THEOREM GIVES THREE OPERATIONS THAT TRANSFORM A LINEAR SYSTEM INTO AN EQUIVALENT SYSTEM AND TOGETHER THEY CAN BE USED TO CONVERT ANY LINEAR SYSTEM TO AN EQUIVALENT SYSTEM IN TRIANGULAR FORM