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Linear Equation
Linear Equation: x,y,z; more generally define a linear equation in the n variables x1, x2, . . . . , xn to be one that can be expressed in the form a1x1+a2x2+. . . +anxn.
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Homogeneous Linear Equation
- Homogeneous Linear Equation:
- b = 0
- [in the variables x1,x2 . . .xn].
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System of linear equations
System of linear equations: A finite set of linear equations or, more briefly, a linear system.
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Variables
- Variables: Unknowns
- ie:
- x, y, z . . . . .
- or
- x1, x2, x3 . . . .
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Ordered n-tuple
- Ordered n-tuple:
- (s1, s2, . . . . , sn)
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N = 2
N = 2: Called an ordered pair, for ordered n-tuple.
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N = 3
N = 3: Called an ordered triple, for ordered n-tuple.
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Consistant
Consistant: Has at least one solution if not infinite.
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Inconsistant
Inconsistant: Has no solutions.
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Parameter
- Parameter: Substitution of a veriable with another (different) variable.
- ie:
- X = t
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Parameter equations
Parameter equations: All the variables are substituted with a different variable (one not used yet).
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Augmented Matrix
Augmented Matrix: Just putting the equations into matrix form.
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Algebraic Operations
- Algerbraic Operations:
- 1. Multiply row by non-0 constant.
- 2. Interchange 2 rows.
- 3. Add a constant times 1 row to another.
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Unique solution
Unique solution: One solution.
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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(B-C)=AB-AC
- g)(B-C)A=BA-CA
- h)a(B+C)=aB+aC
- i)a(B-C)=aB-aC
- j)(a+b)C=aC+bC
- k)(a-b)C=aC-bC
- l)a(bC)=(ab)C
- m)a(BC)=aB(C)=B(aC)
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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)A-0=A
- c)A-A=A+(-A)=0
- d)0A=0
- e)If cA=0, then c=0 or A=0
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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 In.
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