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 Dependent Variable

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# Linear Systems of Equations

Outline Preliminaries Introductions. Sign -up list. Syllabus. Homework 1. Course Overview. Linear Systems of Equations.

Key Points on Syllabus Quizzes every Tuesday. Homeworks due begining of class. NO LATE HOMEWORKS. Late projects are accepted with 20% penalty. NO MAKEUP EXAMS. If you miss an exam, or if you do badly, the weight of that exam will be added to the final .

Homework 1

Due Tuesday, Aug. 30, at beginning of class. Section 1.1: 4,6,8,10,12,14,16,18,20, 23, 24, 30, 33,34 Section 1.2: 2,4,8,10,12, 15, 18, 20, 21,22, 24, 26, 29,
31, 33

Course Overview Dealing with many variables and many equations. Linearity. Abstraction and Proofs.

Linear Equations

A linear equation in the variables is an equation
that can be written in the form Examples: Not Linear: Systems of Linear Equations

An m × n system of linear equations has the form:  Note: each equation involves the same variables, . A solution of the system is a list of numbers ( ) that makes each equation
true when the values are substituted for , respectively.

Example: The system has solution .

A Geometric View

Exercises: Graph the solution set of the equation . (Question: why does it make
sense to call this a linear equation?). Graph the line defined by the equation . Where do these two lines intersect? Geometry 2

But this doesn’t always work: Solutions of Linear Systems

A system of linear equations has either
1. No solution (inconsistent).
2. Exactly one solution. (consistent).
3. Infinitely many solutions. (also consistent).

Matrix Notation

Since all the equations in a linear system involve the same variables, we can economize by
writing only the coefficients (not the variables) in a compact form called the Augmented
matrix: Example: Solving Linear Systems Add (-3) times equation 1 to equation
2. Add equation 2 to equation 1.   Add (-3) times equation 1 to equation 2. Add equation 2 to equation 1. Elementary Row Operations

1. (Replacement) Add c times row j to row i (replacing row
i, and leaving row j unchanged). 2. (Interchange) Interchange row i and row j. 3. (Scaling) Multiply row i by a constant c ≠ 0. KEY FACT: Elementary row operations do not alter the
solution set.

Gauss- Jordan Elimination Apply a sequence of elementary row operations to get the matrix into a form that is
trivial to solve. Example: A series of elementary row operations yields the following transformation The righthand matrix corresponds to the system of equations:  Trivial to solve!! The above matrix is in a special form called the reduced row echelon form.

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