Math 290 Linear Modeling for Engineers Instructor: Minnie Catral Text: Linear Algebra and Its Applications, 3rd ed. Update, by David Lay Have you ever used Matlab as a computing tool? 1. Yes 2. No 2 1.1 Systems of Equations System of 2 equations in 2 unknowns: Three possibilities: 1. The lines intersect. 2. The lines coincide. 3. The lines are parallel. 3 More generally: System of m equations in n unknowns A system is consistent if it has at least one solution. Otherwise, it is said to be inconsistent. 4 FACT: A system of linear equations has either (1) exactly one solution or (2) infinitely many solutions or (3) no solutions. The solution set is the set of all possible solutions of a linear system. Two systems are equivalent if they have the same solution set. STRATEGY FOR SOLVING A SYSTEM: Replace a system with an equivalent system that is easier to solve. 5 • EXAMPLE: 6 Matrix Notation 7 Elementary Row Operations: 1. (Interchange) Interchange two rows. 2. (Scaling) Multiply a row by a nonzero constant. 3. (Replacement) Replace a row by the sum of itself and a multiple of another row. Row equivalent Matrices: Two matrices where one matrix can be transformed into the other matrix by a sequence of elementary row operations. Notation: Fact about Row Equivalence: If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. 8 • EXAMPLE 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 and only one solution?) 10 • EXAMPLE: Is this system consistent? 11 • EXAMPLE. For what value(s) of h will the following system be consistent? 12 Determine the value of h such that the matrix is the augmented matrix of a consistent system. 1. 2. 3. 4. 13 1.2 Row Reduction and Echelon Forms A matrix is in echelon form (or row echelon form) if 1. All nonzero rows are above any rows of all zeros. 2. 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. 3. All entries in a column below a leading entry are zero. 14 Examples. Determine which matrix is in echelon form. (a) (b) (c) 15 A matrix is in echelon form (or row echelon form) if 1. All nonzero rows are above any rows of all zeros. 2. 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. 3. All entries in a column below a leading entry are zero. A matrix is in reduced echelon form (or row reduced echelon form) if in addition to conditions 1,2,3 above: 4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry in its column. 16 Example. is in echelon form but not in reduced echelon form. 17 Uniqueness of the Reduced Echelon Form: Each matrix is row equivalent to one and only one reduced echelon matrix. 18 Important Terms: • pivot position: a position of a leading entry in an echelon form of the matrix • pivot: a nonzero number that either is used in a pivot position to create 0’s or is changed into a leading 1, which in turn is used to create 0’s. • pivot column: a column that contains a pivot position. 19 Example. 20 Solutions of Linear Systems • basic variable: any variable that corresponds to a pivot column in the augmented matrix of a system • free variable: all non-basic variables Example. Find the general solution of the linear system whose augmented matrix is given below: 21.