slides01 - Introduction to systems of linear equations...

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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 x 1 ,...,x n is an equation that can be written as a 1 x 1 + a 2 x 2 + + a n x n = b. Example 2. Which of the following equations are linear? 4 x 1 5 x 2 +2= x 1 x 2 =2( 6 x 1 )+ x 3 4 x 1 6 x 2 = x 1 x 2 x 2 =2 x 1 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, x 1 ,x 2 ,...,x n . A solution of a linear system is 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. Example 4. (Two equations in two variables) In each case, sketch the set of all solutions. x 1 + x 2 = 1 x 1 + x 2 = 0 x 1 2 x 2 = 3 2 x 1 4 x 2 = 8 2 x 1 + x 2 = 1 4 x 1 2 x 2 = 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 [email protected] 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: x 1 + x 2 = 1 x 1 + x 2 = 0 R 2 R 2+ R 1 x 1 + x 2 = 1 2 x 2 = 1 Once in this triangular form, we find the solutions by back-substitution : x 2 =1/2 , x 1 = Example 9. The same approach works for more complicated systems. x 1 2 x 2 + x 3 = 0 2 x 2 8 x 3 = 8 4 x 1 + 5 x 2 + 9 x 3 = 9 R 3 R 3+4 R 1 x 1 2 x 2 + x 3 = 0 2 x 2 8 x 3 = 8 3 x 2 + 13 x 3 = 9 R 3 R 3+ 3 2 R 2 x 1 2 x 2 + x 3 = 0 2 x 2 8 x 3 = 8 x 3 = 3 By back-substitution: x 3 =3 , x 2 = , x 1 = It is always a good idea to check our answer. Let us check that ( 29 , 16 , 3) indeed solves the original system: x 1 2 x 2 + x 3 = 0 2 x 2 8 x 3 = 8 4 x 1 + 5 x 2 + 9 x 3 = 9 Armin Straub [email protected] 2
Matrix notation x 1 2 x 2 = 1 x 1 + 3 x 2 = 3 bracketleftbigg bracketrightbigg (coefficient matrix) bracketleftbigg bracketrightbigg (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.

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