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Linear Equation
Linear Equation: x,y,z; more generally define a linear equation in the n variables x_{1}, x_{2}, . . . . , x_{n} to be one that can be expressed in the form a_{1}x_{1}+a_{2}x_{2}+. . . +a_{n}x_{n}.

Homogeneous Linear Equation
 Homogeneous Linear Equation:
 b = 0
 [in the variables x_{1},x_{2} . . .x_{n}].

System of linear equations
System of linear equations: A finite set of linear equations or, more briefly, a linear system.

Variables
 Variables: Unknowns
 ie:
 x, y, z . . . . .
 or
 x_{1}, x_{2}, x_{3} . . . .

Ordered ntuple
 Ordered ntuple:
 (s_{1}, s_{2}, . . . . , s_{n})

N = 2
N = 2: Called an ordered pair, for ordered ntuple.

N = 3
N = 3: Called an ordered triple, for ordered ntuple.

Consistant
Consistant: Has at least one solution if not infinite.

Inconsistant
Inconsistant: Has no solutions.

Parameter
 Parameter: Substitution of a veriable with another (different) variable.
 ie:
 X = t

Parameter equations
Parameter equations: All the variables are substituted with a different variable (one not used yet).

Augmented Matrix
Augmented Matrix: Just putting the equations into matrix form.

Algebraic Operations
 Algerbraic Operations:
 1. Multiply row by non0 constant.
 2. Interchange 2 rows.
 3. Add a constant times 1 row to another.

Unique solution
Unique solution: One solution.

Theorem 1.4.1
 Theorem 1.4.1:
 a)A+B=B+A _{Communatative Law addition}
 b)A+(B+C)=(A+B)+C _{Associative Law add}
 c)A(BC)=(AB)C _{Associative Law Multi}
 d)A(B+C)=AB+AC _{left distributive Multi}
 e)(B+C)A=BA+CA _{Right distributive Multi}
 f)A(BC)=ABAC
 g)(BC)A=BACA
 h)a(B+C)=aB+aC
 i)a(BC)=aBaC
 j)(a+b)C=aC+bC
 k)(ab)C=aCbC
 l)a(bC)=(ab)C
 m)a(BC)=aB(C)=B(aC)

Theorem 1.4.2
 Theorem 1.4.2: If c is a scalar, and if the sizes of the matrix are such that the operations can be performed, then:
 a) A+0=0+A=A
 b)A0=A
 c)AA=A+(A)=0
 d)0A=0
 e)If cA=0, then c=0 or A=0

Theorem 1.4.3
Theorem 1.4.3: If R is the reduced row echelon form of an nxn matrix A, then either R has a row of zeros or R is the Identity Matrix I_{n}._{}

