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Algebra
Arabic, "aljabr"
completion and balancing

Linear
"line"
introduces a geometric component


Elementary Row Operations
1.) Interchange Rows
2.) Add a multiple of one row to another
3.) Scale a row by a nonzer0 factor

Polynomial
sum of many monomials
 degree is highest exponent
degree one is linear

Free Variables
Columns without leading 1's
 if consistent, has many solutions

Pivot Variables/Leading Variables/Basic Variables
leading 1
 where to start when reducing rows

Pivot Column
Has a leading 1

Pivot Position
used to eliminate nonzero numbers
same positions in first and last matrices (even if different numbers)

Row Reduction
  start with a matrix, end with a solution set
  "Gaussian Elimination"

Row Echelon Form
Get leading 1's in all the rows

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

Augmented Matrix
Matrix with "equal's" column
 RHS and LHS

Coefficient Matrix
Only has coefficients of variables
LHS
 doesn't include "equal's" column
 in a reallife situations, coefficients won't change, but numbers in the equal's column might.

Consistent
has at least one solution
(can have infinitely many solutions)

Inconsistent
no solution
(like with parallel lines)

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

Row Equivalence
can go from one to another with row operations

Solving a System
determine consistency, then find parameters

Overdetermined
More equations than variables
doesn't tell us much

Underdetermined
More variables than equations
  if there are solutions, there are infinitely many
 > free variables

