Linear Algebra Definitions

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Linear Algebra Definitions
2013-01-18 20:08:21
Linear Algebra Definitions

Sections 1.1 through 1.3
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  1. Algebra
    Arabic, "al-jabr"

    completion and balancing
  2. Linear

    introduces a geometric component
  3. Geometry
    Earth + Measure
  4. Elementary Row Operations
    1.) Interchange Rows

    2.) Add a multiple of one row to another

    3.) Scale a row by a nonzer0 factor
  5. Polynomial
    sum of many monomials

    - degree is highest exponent

    -degree one is linear
  6. Free Variables
    Columns without leading 1's

    - if consistent, has many solutions
  7. Pivot Variables/Leading Variables/Basic Variables
    leading 1

    - where to start when reducing rows
  8. Pivot Column
    Has a leading 1
  9. Pivot Position
    used to eliminate nonzero numbers

    -same positions in first and last matrices (even if different numbers)
  10. Row Reduction
    • Algorithm
    • Procedure

    • - start with a matrix, end with a solution set
    • - "Gaussian Elimination"
  11. Row Echelon Form
    Get leading 1's in all the rows
  12. Reduced Row Echelon Form
    a.) Nonzero rows above zero rows

    b.) Leading entries in higher rows is to the right

    c.) All entries below a leading entry are zero

    d.) Leading entries in nonzero rows are 1's

    e.) All entries above a leading entry are 0
  13. Augmented Matrix
    Matrix with "equal's" column

    - RHS and LHS
  14. Coefficient Matrix
    Only has coefficients of variables


    - doesn't include "equal's" column

    - in a real-life situations, coefficients won't change, but numbers in the equal's column might.
  15. Consistent
    has at least one solution

    (can have infinitely many solutions)
  16. Inconsistent
    no solution

    (like with parallel lines)
  17. Existence and Uniqueness of solutions

    How does RREF reflect existence and uniqueness of solutions?
    • Existence:
    • - 0 =/ 1
    •       >no solution exists
    •       > Inconsistent
    • - free variables
    •       > infinitely many solutions exist

    • Uniqueness:
    • - no free variables means solution is unique
  18. Row Equivalence
    can go from one to another with row operations
  19. Solving a System
    determine consistency, then find parameters
  20. Overdetermined
    More equations than variables

    doesn't tell us much
  21. Underdetermined
    More variables than equations

    • - if there are solutions, there are infinitely many
    •      > free variables