Introduction to systems of linear equations These slides are based on Section 1 in Linear Algebra and its Applications by David C Lay Definition 1 A linear equation in the variables x1 xn is an equation that can be written as a 1 x 1 a 2x 2 anxn b Example 2 Which of the following equations are linear 4x1 5x2 2 x1 x2 2 6 x1 x3 4x1 6x2 x1x2 x2 2 x1 7 Definition 3 A system of linear equations or a linear system is a collection of one or more linear equations involving the same set of variables say x1 x2 xn A solution of a linear system is a list s1 s2 sn of numbers that makes each equation in the system true when the values s1 s2 sn are substituted for x1 x2 xn respectively Example 4 Two equations in two variables In each case sketch the set of all solutions x1 x2 1 x1 x2 0 x1 2x2 3 2x1 4x2 8 2x1 x2 1 4x1 2x2 2 Theorem 5 A linear system has either no solution or one unique solution or infinitely many solutions Definition 6 A system is consistent if a solution exists Armin Straub astraub illinois edu 1 How to solve systems of linear equations Strategy replace system with an equivalent system which is easier to solve Definition 7 Linear systems are equivalent if they have the same set of solutions Example 8 To solve the first system from the previous example x1 x2 1 x1 x2 0 R2 R2 R1 x1 x2 1 2x2 1 Once in this triangular form we find the solutions by back substitution x2 1 2 x1 Example 9 The same approach works for more complicated systems x1 2x2 x3 0 2x2 8x3 8 4x1 5x2 9x3 9 x1 2x2 x3 0 2x2 8x3 8 3x2 13x3 9 R3 R3 4R1 3 R3 R3 2 R2 x1 2x2 x3 0 2x2 8x3 8 x3 3 By back substitution x3 3 x2 x1 It is always a good idea to check our answer Let us check that 29 16 3 indeed solves the original system x1 2x2 x3 0 2x2 8x3 8 4x1 5x2 9x3 9 Armin Straub astraub illinois edu 2 Matrix notation x1 2x2 1 x1 3x2 3 coefficient matrix augmented matrix Definition 10 An elementary row operation is one of the following replacement Add one row to a multiple of another row interchange Interchange two rows scaling Multiply all entries in a row by a nonzero constant Definition 11 Two matrices are row equivalent if one matrix can be transformed into the other matrix by a sequence of elementary row operations Theorem 12 If the augmented matrices of two linear systems are row equivalent then the two systems have the same solution set Example 13 Here is the previous example in matrix notation x1 2x2 x3 0 1 2 1 0 0 2x2 8x3 8 8 2 8 4x1 5x2 9x3 9 4 5 9 9 x1 2x2 x3 0 2x2 8x3 8 3x2 13x3 9 x1 2x2 x3 0 2x2 8x3 8 x3 3 1 2 1 0 8 0 2 8 0 3 13 9 1 2 1 0 2 8 0 0 1 Instead of back substitution we can continue with x1 2x2 3 1 2 2x2 32 0 2 x3 3 0 0 x1 x2 x3 29 16 3 1 0 0 0 1 0 R3 R3 4R1 3 R3 R3 2 R2 0 8 3 row operations 0 3 0 32 1 3 0 0 1 29 16 3 We again find the solution x1 x2 x3 29 16 3 Armin Straub astraub illinois edu 3 Row reduction and echelon forms Definition 14 A matrix is in echelon form or row echelon form if 1 Each leading entry i e leftmost nonzero entry of a row is in a column to the right of the leading entry of the row above it 2 All entries in a column below a leading entry are zero 3 All nonzero rows are above any rows of all zeros Example 15 Here is a representative 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 matrix in echelon form 0 0 0 0 0 0 0 0 0 0 0 0 0 0 stands for any value and for any nonzero value Example 0 a 0 0 0 b 0 0 0 c 0 0 d 16 Are the following matrices in echelon form 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Definition 17 A leading entry in an echelon form is called a pivot Definition 18 A matrix is in reduced echelon form if in addition to being in echelon form it also satisfies 4 Each pivot is 1 5 Each pivot is the only nonzero entry in its column Armin Straub astraub illinois edu 4 Example 19 Our initial matrix in echelon form put into reduced echelon form 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Locate the pivots Example 20 Are the following matrices in reduced echelon form 0 1 0 0 0 0 0 0 0 1 0 0 0 a 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 5 0 7 0 2 4 0 6 b 0 0 0 5 0 0 0 0 0 0 1 0 2 3 2 24 c 0 1 2 2 0 7 0 0 0 0 1 4 Theorem 21 Uniqueness of the reduced echelon form Each matrix is row equivalent to one and only one reduced echelon matrix Question Is the same statement true for the echelon form Example 22 Row reduce to echelon form often called Gaussian elimination and then to reduced echelon form often called Gauss Jordan elimination 0 3 6 6 4 5 3 7 8 5 8 9 3 9 12 9 6 15 Solution Hence the reduced echelon form is 1 0 2 3 0 24 0 1 2 2 0 7 0 0 0 0 1 4 Armin Straub astraub illinois edu 5 Solution of linear systems via row reduction After row reduction to echelon form we can easily solve a linear system especially after reduction to reduced echelon form Example 23 1 6 0 3 0 0 0 0 1 8 0 5 0 0 0 0 1 7 x1 6x2 x3 3x4 8x4 x5 0 5 7 The pivots are located in columns 1 3 5 The corresponding variables x1 x3 x5 are called pivot …
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