Math 2331 Linear Algebra

Transcription

1 1.1 Linear System Math 2331 Linear Algebra 1.1 Systems of Linear Equations Shang-Huan Chiu Department of Mathematics, University of Houston math.uh.edu/ schiu/ Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

2 1.1 Systems of Linear Equations Basic Fact on Solution of a Linear System Example: Two Equations in Two Variables Example: Three Equations in Three Variables Consistency Equivalent Systems Strategy for Solving a Linear System Matrix Notation Solving a System in Matrix Form by Row Eliminations Elementary Row Operations Row Eliminations to a Triangular Form Row Eliminations to a Diagonal Form Two Fundamental Questions Existence Uniqueness Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

3 Linear Equation A Linear Equation a 1 x 1 + a 2 x a n x n = b Examples (Linear) 4x 1 5x = x 1 and x 2 = 2( 6 x 1 ) + x 3 rearranged rearranged 3x 1 5x 2 = 2 2x 1 + x 2 x 3 = 2 6 Examples (Not Linear) 4x 1 6x 2 = x 1 x 2 and x 2 = 2 x 1 7 Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

4 Linear System A solution of a System of Linear Equations A list (s 1, s 2,..., s n ) of numbers that makes each equation in the system true when the values s 1, s 2,..., s n are substituted for x 1, x 2,..., x n, respectively. Examples (Two Equations in Two Variables) Each equation determines a line in 2-space. x 1 + x 2 = 10 x 1 + x 2 = 0 x 1 2x 2 = 3 2x 1 4x 2 = 8 one unique solution no solution Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

5 Basic Fact on Solution Basic Fact on Solution of a Linear System 1 exactly one solution (consistent) or 2 infinitely many solutions (consistent) or 3 no solution (inconsistent). Examples (Two Equ. Two Var.) x 1 + x 2 = 3 2x 1 2x 2 = 6 infinitely many solutions Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

6 Basic Fact on Solution (cont.) Examples (Three Equations in Three Variables) Each equation determines a plane in 3-space. i) The planes intersect in ii) There is not point in common one point. (one solution) to all three planes. (no solution) Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

7 Equivalent Systems Solution Set of a Linear System The set of all possible solutions of a linear system. Examples (Two Equ. Two Var.) x 1 2x 2 = 1 x 1 + 3x 2 = 3 Equivalent Systems Two linear systems with the same solution set. STRATEGY FOR SOLVING A SYSTEM Replace one system with an equivalent system that is easier to solve. x 1 2x 2 = 1 x 2 = 2 x 1 = 3 x 2 = 2 Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

8 Equivalent Systems (cont.) Examples (Two Equ. in Two Var. (cont.)) x 1 2x 2 = 1 x 1 + 3x 2 = 3 x 1 2x 2 = 1 x 2 = 2 Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

9 Equivalent Systems (cont.) Examples (Two Equ. in Two Var. (cont.)) x 1 2x 2 = 1 x 2 = 2 x 1 = 3 x 2 = 2 Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

10 Matrix Notation Example (Coefficient Matrix: Two Row and Two Columns) x 1 2x 2 = 1 [ 1 ] 2 x 1 + 3x 2 = (coefficient matrix) Example (Augmented Matrix: Two Row and Three Columns) x 1 2x 2 = 1 [ ] x 1 + 3x 2 = (augmented matrix) Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

11 Solving a Linear System Example Solving a System in Matrix Form[ x 1 2x 2 = ] x 1 + 3x 2 = (augmented matrix) x 1 2x 2 = 1 [ ] x 2 = x 1 = 3 x 2 = 2 [ ] Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

12 Row Operations 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 can be transformed into the other matrix by a sequence of elementary row operations. Fact about Row Equivalence If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

13 Solving a System by Row Eliminations: Example Example (Row Eliminations to a Triangular Form) x 1 2x 2 + x 3 = 0 2x 2 8x 3 = 8 4x 1 + 5x 2 + 9x 3 = 9 x 1 2x 2 + x 3 = 0 2x 2 8x 3 = 8 3x x 3 = 9 x 1 2x 2 + x 3 = 0 x 2 4x 3 = 4 3x x 3 = 9 x 1 2x 2 + x 3 = 0 x 2 4x 3 = 4 x 3 = Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

14 Solving a System by Row Eliminations: Example (cont.) Example (Row Eliminations to a Diagonal Form) x 1 2x 2 + x 3 = 0 x 2 4x 3 = 4 x 3 = 3 x 1 2x 2 = 3 x 2 = 16 x 3 = 3 x 1 = 29 x 2 = 16 x 3 = 3 Solution: (29, 16, 3) Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

15 Solving a System by Row Eliminations: Example (cont.) Example (Check the Answer) Is (29, 16, 3) a solution of the original system? x 1 2x 2 + x 3 = 0 2x 2 8x 3 = 8 4x 1 + 5x 2 + 9x 3 = 9 (29) 2(16)+ (3) = = 0 2(16) 8(3) = = 8 4(29) + 5(16) + 9(3) = = 9 Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

16 Existence and Uniqueness 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 one solution?) Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

17 Existence: Examples Example (Is this system consistent?) x 1 2x 2 + x 3 = 0 2x 2 8x 3 = 8 4x 1 + 5x 2 + 9x 3 = 9 In the last example, this system was reduced to the triangular form: x 1 2x 2 + x 3 = x 2 4x 3 = x 3 = This is sufficient to see that the system is consistent and unique. Why? Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

18 Existence: Examples (cont.) Example (Is this system consistent?) 3x 2 6x 3 = 8 x 1 2x 2 + 3x 3 = 1 5x 1 7x 2 + 9x 3 = Solution: Row operations produce: Equation notation of triangular form: x 1 2x 2 + 3x 3 = 1 3x 2 6x 3 = 8 0x 3 = 3 Never true The original system is inconsistent! Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19

19 Existence: Examples (cont.) Example (For what values of h will the system be consistent?) 3x 1 9x 2 = 4 2x 1 + 6x 2 = h Solution: Reduce to triangular form. [ h ] [ h ] [ h ] The second equation is 0x 1 + 0x 2 = h System is consistent only if h = 0 or h = 3. Shang-Huan Chiu, University of Houston Math 2331, Linear Algebra Fall, / 19