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Lect1-03-web

# Lect1-03-web - MATH 304 Fall 2011 Linear Algebra MATH 304...

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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra MATH 304 Linear Algebra Lecture 3: Some applications of systems of linear equations. Matrix algebra. Homework assignment Homework assignment 1 (due Thursday, September 8): Section 1.1: 1b, 1d, 3a, 3d, 6e, 6h Section 1.2: 1a, 1c, 1e, 1g, 3a, 3c, 5i, 9, 22b Additionally! Section 1.2 : 6b, 6c How to solve a system of linear equations • Order the variables • Write down the augmented matrix of the system • Convert the matrix to row echelon form • Check for consistency • Convert the matrix to reduced row echelon form • Write down the system corresponding to the reduced row echelon form • Determine leading and free variables • Rewrite the system so that the leading variables are on the left while everything else is on the right • Write down the general solution in parametric form System with a parameter y + 3 z = 0 x + y − 2 z = 0 x + 2 y + az = 0 ( a ∈ R ) The system is homogeneous (all right-hand sides are zeros). Therefore it is consistent ( x = y = z = 0 is a solution). Augmented matrix: 0 1 3 1 1 − 2 1 2 a Since the 1st row cannot serve as a pivotal one, we interchange it with the 2nd row: 0 1 3 1 1 − 2 1 2 a → 1 1 − 2 0 1 3 1 2 a Now we can start the elimination. First subtract the 1st row from the 3rd row: 1 1 − 2 0 1 3 1 2 a → 1 1 − 2 0 1 3 0 1 a + 2 The 2nd row is our new pivotal row. Subtract the 2nd row from the 3rd row: 1 1 − 2 0 1 3 0 1 a + 2 → 1 1 − 2 0 1 3 0 0 a − 1 At this point row reduction splits into two cases. Case 1: a negationslash = 1. In this case, multiply the 3rd row by ( a − 1) − 1 : 1 1 − 2 0 1 3 0 0 a − 1 → 1 1 − 2 1 3 1 The matrix is converted into row echelon form. We proceed towards reduced row echelon form. Subtract 3 times the 3rd row from the 2nd row: 1 1 − 2 0 1 3 0 0 1 → 1 1 − 2 0 1 0 0 1 Add 2 times the 3rd row to the 1st row: 1 1 − 2 0 1 0 0 1 → 1 1 0 0 1 0 0 0 1 Finally, subtract the 2nd row from the 1st row:...
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Lect1-03-web - MATH 304 Fall 2011 Linear Algebra MATH 304...

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