Section 1.1: Systems of Linear Equations Two Linear Equations in Two Unknowns Recall that the equation of a line in 2D can be written in standard form: a 1 x 1 + a 2 x 2 = b. Definition. A 2 2 system of linear equations has two equations and two unknowns (x 1, x 2 ): a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 The a ij s and b i s are real numbers and the x i s are variables. A solution to a 2 2 linear system is a 2-tuple (x 1, x 2 ) of numbers that satisfies both equations in the system. The set of all solutions is called the solution set. A system is called consistent if it has at least one solution, or inconsistent otherwise. Three situations can occur: the two equations have lines (a) intersecting at a single point, (b) parallel and intersecting nowhere, or (c) parallel and are the same line. (a) (b) (c) x 1 + x 2 = 5 x 1 + x 2 = 5 x 1 + x 2 = 5 x 1 x 2 = 5 x 1 + x 2 = 0 x 1 x 2 = 5 consistent inconsistent consistent solution(s): (5, 0) solution(s): none solution(s): {(α, 5 α) : α R} In general, these are the only three situations that can occur. If the system is consistent, it has either exactly one solution or an infinite number of solutions.
M309 Notes, R.G. Lynch, Texas A&M Section 1.1: Systems of Linear Equations Page 2 of 6 General Linear Systems Nothing is special about the case of two equations with two unknowns. We can generalize all of these concepts to m equations with n unknowns. Definition. A linear equation with n unknowns (that is, in n dimensions) is of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a i, b R are numbers are the x i s are variables. A linear system with m equations in n unknowns is of the form a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m A solution of the system is an n-tuple (x 1, x 2,..., x n ) that satisfies all m equations. Consistency is defined in the same way as 2 2 systems and again, we can only have exactly one solution, an infinite number of solutions, or no solutions. Example. Let s examine the solution sets of the following linear systems (a) 3x 1 + 2x 2 x 3 = 2 x 2 = 3 2x 3 = 4. (b) 3x 1 + 2x 2 x 3 = 2 3x 1 x 2 + x 3 = 5 3x 1 + 2x 2 + x 3 = 2 Definition. Two systems of equations involving the same variables are equivalent if they have the same solution set. There are three operations that can be used on a system to obtain an equivalent system: I. The order of the equations can be changed. II. Both sides of an equation may be multiplied by the same nonzero real number. III. A multiple of one equation can be added/subtracted to another.
M309 Notes, R.G. Lynch, Texas A&M Section 1.1: Systems of Linear Equations Page 3 of 6 n n Systems Definition. A system is said to be in strict triangular form if, in the kth equation, the coefficients of the first k 1 variables are all zero and the coefficient of x k is nonzero. If an n n linear system can be changed to strict triangular form, then it has a unique solution. Example. Solve the following systems. We use a method called back substitution. (a) 3x 1 + 2x 2 + x 3 = 1 x 2 x 3 = 2 2x 3 = 4 (b) 2x 1 x 2 + 3x 3 2x 4 = 1 x 2 2x 3 + 3x 4 = 2 4x 3 + 3x 4 = 3 4x 4 = 4
M309 Notes, R.G. Lynch, Texas A&M Section 1.1: Systems of Linear Equations Page 4 of 6 (c) x 1 + 2x 2 + x 3 = 3 3x 1 x 2 3x 3 = 1 2x 1 + 3x 2 + x 3 = 4 Definition. Looking at the last example, we can associate with it a 3 3 array of numbers who entries are the coefficients of the x i s: 1 2 1 3 1 3 2 3 1 called the coefficient matrix of the system. A matrix is simply a rectangular array of numbers. A matrix with m rows and n columns is said to be m n and if m = n it is called square. Definition. We can also attach to the coefficient matrix the another column with the b i s in it: 1 2 1 3 3 1 3 1 2 3 1 4 This is called an augmented matrix. We can actually attach any two matrices with the same number of rows in this way, not just a square one and another column. That is, if A is an m r matrix and B is an m s matrix, then the augmented matrix (A B) is a m (r + s) matrix. Definition. Since the augmented matrix of a linear system is just a convenient way to represent a linear system of equations, we can apply the same three operations to it. These are called Elementary Row Operations: I. Interchange two rows. II. Multiple a row by a nonzero real number. III. Replace a row by its sum with a multiple of another row. We will typically use these to get the augmented matrix into strictly triangular form and then use back substitution to find the solution(s) to the system if they exist.
M309 Notes, R.G. Lynch, Texas A&M Section 1.1: Systems of Linear Equations Page 5 of 6 Example. Write the augmented matrix of the system and use it to solve. x 2 x 3 + x 4 = 0 x 1 + x 2 + x 3 + x 4 = 6 2x 1 + 4x 2 + x 3 2x 4 = 1 3x 1 + x 2 2x 3 + 2x 4 = 3
M309 Notes, R.G. Lynch, Texas A&M Section 1.1: Systems of Linear Equations Page 6 of 6 Example. Write out the system of equations that corresponds to augmented matrix 2 1 4 1 4 2 3 4 5 2 6 1 and then use this matrix to solve the system.