Section 1.1: Systems of Linear Equations A linear equation: a1x1 ...

Section 1.1: Systems of Linear EquationsA linear equation:a 1 x 1 a 2 x 2 a n x n bEXAMPLE:4x 1 5x 2 2 x 1 and x 2 2 6 x 1 x 3rearrangedrearranged3x 1 5x 2 2 2x 1 x 2 x 3 2 6Not linear:4x 1 6x 2 x 1 x 2 and x 2 2 x 1 7A system of linear equations (or a linear system):A collection of one or more linear equations involving the sameset of variables, say, x 1 ,x 2 ,...,x n .A solution of a linear system:A list s 1 ,s 2 ,...,s n of numbers that makes each equation in thesystem true when the values s 1 ,s 2 ,...,s n are substituted forx 1 ,x 2 ,...,x n , respectively.1

EXAMPLE Two equations in two variables:x 1 x 2 10x 1 2x 2 3x 1 x 2 02x 1 4x 2 8x 2−1 1 2 3 4 5 x 11086422 4 6 8 10 x 14321−1−2x 2one unique solutionno solutionx 1 x 2 32x 1 2x 2 6x 24321−1−11 2 3 4 5 x 1−2infinitely many solutionsBASIC FACT: A system of linear equations has either(i) exactly one solution (consistent) or(ii) infinitely many solutions (consistent) or(iii) no solution (inconsistent).2

EXAMPLE: Three equations in three variables. Each equationdetermines a plane in 3-space.i) The planes intersect in ii) The planes intersect in oneone point. (one solution) line. (infinitely many solutions)iii) There is not point in commonto all three planes. (no solution)3

The solution set: The set of all possible solutions of a linear system.Equivalent systems: Two linear systems with the same solution set.STRATEGY FOR SOLVING A SYSTEM: Replace one system with an equivalent system that iseasier to solve.EXAMPLE:x 1 2x 2 1x 1 3x 2 3x 1 2x 2 1x 2 2x 1 3x 2 24

x 2 x 2−10 −5 5 10 x 14422−10 −5 5 10 x 1−2−2−4−4x 1 2x 2 1x 1 3x 2 3x 1 2x 2 1x 2 242−10 −5 5 10−2−4x 1 3x 2 25

Matrix Notationx 1 2x 2 1x 1 3x 2 31 21 3(coefficient matrix)x 1 2x 2 1x 1 3x 2 31 2 11 3 3(augmented matrix)x 1 2x 2 1x 2 21 2 10 1 2x 1 3x 2 21 0 30 1 26

Elementary Row Operations:1. (Replacement) Add one row to a multiple of another row.2. (Interchange) Interchange two rows.3. (Scaling) Multiply all entries in a row by a nonzero constant.Row equivalent matrices: Two matrices where one matrix canbe transformed into the other matrix by a sequence of elementaryrow operations.Fact about Row Equivalence: If the augmented matrices of twolinear systems are row equivalent, then the two systems have thesame solution set.7

EXAMPLE:x 1 2x 2 x 3 02x 2 8x 3 84x 1 5x 2 9x 3 91 2 1 00 2 8 84 5 9 9x 1 2x 2 x 3 02x 2 8x 3 8 3x 2 13x 3 91 2 1 00 2 8 80 3 13 9x 1 2x 2 x 3 0x 2 4x 3 4 3x 2 13x 3 91 2 1 00 1 4 40 3 13 9x 1 2x 2 x 3 0x 2 4x 3 4x 3 31 2 1 00 1 4 40 0 1 3x 1 2x 2 3x 2 16x 3 31 2 0 30 1 0 160 0 1 38

x 1 29x 2 16x 3 31 0 0 290 1 0 160 0 1 3Solution: 29,16,3Check: Is 29,16,3 a solution of the original system?x 1 2x 2 x 3 02x 2 8x 3 84x 1 5x 2 9x 3 929 216 3 29 32 3 0216 83 32 24 8429 516 93 116 80 27 99

Two Fundamental Questions (Existence and Uniqueness)1) Is the system consistent; (i.e. does a solution exist?)2) If a solution exists, is it unique? (i.e. is there one & only onesolution?)EXAMPLE: Is this system consistent?x 1 2x 2 x 3 02x 2 8x 3 84x 1 5x 2 9x 3 9In the last example, this system was reduced to the triangularform:x 1 2x 2 x 3 0x 2 4x 3 4x 3 31 2 1 00 1 4 40 0 1 3This is sufficient to see that the system is consistent and unique.Why?10

EXAMPLE: Is this system consistent?3x 2 6x 3 8x 1 2x 2 3x 3 15x 1 7x 2 9x 3 00 3 6 81 2 3 15 7 9 0Solution: Row operations produce:0 3 6 81 2 3 11 2 3 10 3 6 81 2 3 10 3 6 85 7 9 00 3 6 50 0 0 3Equation notation of triangular form:x 1 2x 2 3x 3 13x 2 6x 3 80x 3 3 Never trueThe original system is inconsistent!11

EXAMPLE: For what values of h will the following system beconsistent?3x 1 9x 2 42x 1 6x 2 hSolution: Reduce to triangular form.3 9 42 6 h1 3432 6 h41 330 0 h 8 3The second equation is 0x 1 0x 2 h 8 3only if h 8 8 0 or h 3. System is consistent3 . 12