Section 1.1: Systems of Linear Equations

Abstract:

We begin with linear equations, and systems of linear equations. The variable names are essentially irrelevant to the solution set, so matrix notation eliminates the need to even give them names!

Given a system, the idea is to replace the system by one that's easier to solve, yet retains the solutions of the original system. This is done by elementary row operations (replacement, interchange, and scaling). Finally a triangular system is obtained, and the solution can be obtained by back-substitution (if a solution exists!).

Geometrically, the solution set of a system of linear equations corresponds to the intersection of linear objects embedded in space. There may be no solution, a unique solution, or an infinite number of solutions.

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• Definition: linear equation

  an equation of the form

  

  where b and the are generally real or complex numbers, and the are variables.

  Definition: system of linear equations

  A set of linear equations which are to be true simultaneously

  Example:

  

  Definition: solution set of a linear system

  the set of all solutions of a linear system

  1. Question: How did we solve that back in High School?
  2. Question: What are the geometric possibilities?
  3. Question: How could we screw up that system so that it wouldn't have a solution?
  4. Question: How can technology help?

• Theorem:

  A system of linear equations has either

  • no solution, or
  • exactly one solution, or
  • infinitely many solutions.

• Definition: matrix

  a rectangular array of numbers (or even more general objects)

  Matrices allow us to eliminate unnecessary stuff! They suppress variable names, which are irrelevant to the solution.

  1. coefficient matrix

     matrix of the coefficients of a linear system

  2. augmented matrix

     matrix of the coefficients of a linear system and an added column containing the right hand side of the linear system

  3. size of a matrix

     indicates the number of rows and the number of columns of a matrix (e.g. 3 by 4 - 3 rows, 4 columns)

• Using matrix notation, we can now solve systems using elementary row operations:
  • The point: Replace a given system by another system with the same solution set that's easier to solve
  • What's legal?
    1. replacement of a row by a sum of the row and multiples of other rows
    2. interchange of rows
    3. scaling of a row by a nonzero constant

    Definition: two matrices are row-equivalent if they can be transformed back and forth using elementary row operations.

  Example: #12, p. 11

  How can technology hurt?

• Fundamental question about the solution's Existence and Uniqueness:
  • Is there at least one solution? (consistent; otherwise inconsistent)
  • Given that a solution exists (existence), is the solution unique (uniqueness)?

  Example: #20, p. 11

• Additional examples: p. 11, #15, 26, 32