Math 2B, College of the Desert

Linear Equation: x,y,z; more generally define a linear equation in the n variables x₁, x₂, . . . . , x_n to be one that can be expressed in the form a₁x₁+a₂x₂+. . . +a_nx_n.

• Homogeneous Linear Equation:
• b = 0
• [in the variables x₁,x₂ . . .x_n].

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

• Variables: Unknowns
• ie:
• x, y, z . . . . .
• or
• x₁, x₂, x₃ . . . .

• Ordered n-tuple:
• (s₁, s₂, . . . . , s_n)

N = 2: Called an ordered pair, for ordered n-tuple.

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.
• ie:
• 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 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)

• 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

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.