Section 1.1: Systems of Linear Equations A linear equation: a 1 x 1 a 2 x 2 a n x n b EXAMPLE: 4x 1 5x 2 2 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 Not linear: 4x 1 6x 2 x 1 x 2 and x 2 2 x 1 7 A system of linear equations (or a linear system): A collection of one or more linear equations involving the same set 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 the system true when the values s 1,s 2,...,s n are substituted for x 1,x 2,...,x n, respectively. 1
EXAMPLE Two equations in two variables: x 1 x 2 10 x 1 2x 2 3 x 1 x 2 0 2x 1 4x 2 8 x 2 10 8 6 4 2 2 4 6 8 10 x 1 3 2 1 1 1 2 3 4 5 x 1 1 2 x 2 4 one unique solution no solution x 1 x 2 3 2x 1 2x 2 6 x 2 4 3 2 1 1 1 1 2 3 4 5 x 1 2 infinitely many solutions BASIC 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 equation determines a plane in 3-space. i) The planes intersect in ii) The planes intersect in one one point. (one solution) line. (infinitely many solutions) iii) There is not point in common to 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 is easier to solve. EXAMPLE: x 1 2x 2 1 x 1 3x 2 3 x 1 2x 2 1 x 2 2 x 1 3 x 2 2 4
x 2 x 2 4 4 2 2 10 5 5 10 x 1 2 10 5 5 10 x 1 2 4 4 x 1 2x 2 1 x 1 3x 2 3 x 1 2x 2 1 x 2 2 4 2 10 5 5 10 2 4 x 1 3 x 2 2 5
Matrix Notation x 1 2x 2 1 x 1 3x 2 3 1 2 1 3 (coefficient matrix) x 1 2x 2 1 x 1 3x 2 3 1 2 1 1 3 3 (augmented matrix) x 1 2x 2 1 x 2 2 1 2 1 0 1 2 x 1 3 x 2 2 1 0 3 0 1 2 6
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. 7
EXAMPLE: x 1 2x 2 x 3 0 2x 2 8x 3 8 4x 1 5x 2 9x 3 9 1 2 1 0 0 2 8 8 4 5 9 9 x 1 2x 2 x 3 0 2x 2 8x 3 8 3x 2 13x 3 9 1 2 1 0 0 2 8 8 0 3 13 9 x 1 2x 2 x 3 0 x 2 4x 3 4 3x 2 13x 3 9 1 2 1 0 0 1 4 4 0 3 13 9 x 1 2x 2 x 3 0 x 2 4x 3 4 x 3 3 1 2 1 0 0 1 4 4 0 0 1 3 x 1 2x 2 3 x 2 16 x 3 3 1 2 0 3 0 1 0 16 0 0 1 3 8
x 1 29 x 2 16 x 3 3 1 0 0 29 0 1 0 16 0 0 1 3 Solution: 29,16,3 Check: 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 29 32 3 0 2 16 8 3 32 24 8 4 29 5 16 9 3 116 80 27 9 9
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?) 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 0 x 2 4x 3 4 x 3 3 1 2 1 0 0 1 4 4 0 0 1 3 This is sufficient to see that the system is consistent and unique. Why? 10
EXAMPLE: Is this system consistent? 3x 2 6x 3 8 x 1 2x 2 3x 3 1 5x 1 7x 2 9x 3 0 0 3 6 8 1 2 3 1 5 7 9 0 Solution: Row operations produce: 0 3 6 8 1 2 3 1 1 2 3 1 0 3 6 8 1 2 3 1 0 3 6 8 5 7 9 0 0 3 6 5 0 0 0 3 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! 11
EXAMPLE: For what values of h will the following system be consistent? 3x 1 9x 2 4 2x 1 6x 2 h Solution: Reduce to triangular form. 3 9 4 2 6 h 1 3 4 3 2 6 h 4 1 3 3 0 0 h 8 3 The second equation is 0x 1 0x 2 h 8 3 only if h 8 8 0 or h 3 3.. System is consistent 12
9 σ 1 V 1 σ 4 0 σ 3 V σ 5 σ 2 V 2 + V Why do we care? Figure 1: A simple resitor network. Example 3 (Resistor Networks). Consider a network of resistors; According to Kirkoff s (conservation of charge) and Ohm s law (V = IR for I = σv where σ = 1/R) we have, given V, that V 1 and V 2 satisfy the equations; σ 1 (V 1 0) + σ 2 (V 1 V ) + σ 3 (V 1 V 2 ) = 0 σ 2 (V 2 0) + σ 3 (V 2 V 1 ) + σ 5 (V 2 V ) = 0.
10 This is equivalent to the system of equations; (σ 1 + σ 2 + σ 3 ) V 1 σ 3 V 2 = σ 2 V σ 3 V 1 + (σ 2 + σ 3 + σ 5 ) V 2 = σ 5 V. We would certainly like to know that these equations have unique solutions!
11 Example 4 (Equilibrium Heat Distribution). A numerical approximation to the heat distribution on as steel plate. The temperature at an interior node is the avergage of its nearest neighbors (Newton s law of cooling!) For example We will get 9 equations in 9 unknowns. t 1 = 1 4 (0 + 20 + t 2 + t 4 ) t 5 = 1 4 (t 2 + t 4 + t 6 + t 8 ).
12 Example 5 (Discrete Heat Equation). If we are not at equilibrium in the above example, then we have to solve the system of linear differential equations; so for example; d ds t k (s) = Avg Temp. of Neighbors t k (s) d ds t 1 (s) = 1 4 (0 + 20 + t 2 (s) + t 4 (s)) t 1 (s) d ds t 5 (s) = 1 4 (t 2 (s) + t 4 (s) + t 6 (s) + t 8 (s)) t 5 (s).
13 k k k m m x 1 x 2 Figure 2: A simple spring and mass system.fs Example 6 (Oscillators = Wave Equations)). Consider the following spring mass system; mẍ 1 (t) = kx 1 (t) + k (x 2 (t) x 1 (t)) = 2kx 1 (t) + kx 2 (t) mẍ 2 (t) = kx 2 (t) k (x 2 (t) x 1 (t)) = kx 1 2kx 2.
14 Example 7 (Markov Chains). Let P = 0 1/2 1/2 1/3 1/3 1/3 1/4 0 3/4 where P ij is the probability of jumping from site i to j. We would like to know what are the dynamics and equilibrium properties of this system.
15 1/3 1/3 1 2 1/2 1/2 1/4 1/3 0 3 3/4 Figure 3: A jump diagram for a Markov chain which we study the long run behavior.
16 1. Differential calculus derivative is the best linear approximation 2. Integral calculus determinants are used to compute areas, volumes, etc. 3. Least squares fitting. 4. Principle component analysis (Singular Value Decomposition) (NETFLIX) 5. Optimization problems linear programming. 6. Google Page Rank see MatLab assignment. 7. Linear control theory how to make a plane fly. 8. Filtering theory finding a signal in the noise 9. Quantum mechanics = matrix mechanics. 10. Computer graphics and rigid body mechanics 11. Image/Data compression 12. Data Encryption algorithms. 13. Tensors; stress, metric, field strengths, etc. used in almost all physical theories.