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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.
Homogeneous Linear Equation
- Homogeneous Linear Equation:
- b = 0
- [in the variables x1,x2 . . .xn].
System of linear equations
System of linear equations: A finite set of linear equations or, more briefly, a linear system.
- Variables: Unknowns
- x, y, z . . . . .
- x1, x2, x3 . . . .
- Ordered n-tuple:
- (s1, s2, . . . . , sn)
N = 2
N = 2: Called an ordered pair, for ordered n-tuple.
N = 3
N = 3: Called an ordered triple, for ordered n-tuple.
Consistant: Has at least one solution if not infinite.
Inconsistant: Has no solutions.
- Parameter: Substitution of a veriable with another (different) variable.
- X = t
Parameter equations: All the variables are substituted with a different variable (one not used yet).
Augmented Matrix: Just putting the equations into matrix form.
- Algerbraic Operations:
- 1. Multiply row by non-0 constant.
- 2. Interchange 2 rows.
- 3. Add a constant times 1 row to another.
Unique solution: One solution.
- Theorem 1.4.1:
- a)A+B=B+A Communatative Law additionb)A+(B+C)=(A+B)+C Associative Law addc)A(BC)=(AB)C Associative Law Multid)A(B+C)=AB+AC left distributive Multie)(B+C)A=BA+CA Right distributive Multif)A(B-C)=AB-AC
- 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
- e)If cA=0, then c=0 or A=0
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.