Math 2B, College of the Desert

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Mattyj1388
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81586
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Math 2B, College of the Desert
Updated:
2011-06-23 18:26:48
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Chapter1 math 2B
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Elementary Linear Algerbra: Howard Anton / Chris Rorres
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  1. 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.
  2. Homogeneous Linear Equation
    • Homogeneous Linear Equation:
    • b = 0
    • [in the variables x1,x2 . . .xn].
  3. System of linear equations
    System of linear equations: A finite set of linear equations or, more briefly, a linear system.
  4. Variables
    • Variables: Unknowns
    • ie:
    • x, y, z . . . . .
    • or
    • x1, x2, x3 . . . .
  5. Ordered n-tuple
    • Ordered n-tuple:
    • (s1, s2, . . . . , sn)
  6. N = 2
    N = 2: Called an ordered pair, for ordered n-tuple.
  7. N = 3
    N = 3: Called an ordered triple, for ordered n-tuple.
  8. Consistant
    Consistant: Has at least one solution if not infinite.
  9. Inconsistant
    Inconsistant: Has no solutions.
  10. Parameter
    • Parameter: Substitution of a veriable with another (different) variable.
    • ie:
    • X = t
  11. Parameter equations
    Parameter equations: All the variables are substituted with a different variable (one not used yet).
  12. Augmented Matrix
    Augmented Matrix: Just putting the equations into matrix form.
  13. Algebraic Operations
    • Algerbraic Operations:
    • 1. Multiply row by non-0 constant.
    • 2. Interchange 2 rows.
    • 3. Add a constant times 1 row to another.
  14. Unique solution
    Unique solution: One solution.
  15. 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)
  16. 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
  17. 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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