Definition of Linear Equation:
Coefficients:
a
and
b
Variables:
x
1
and
x
2
Theorem:
A system of linear equations has a unique solution, infinitely many solutions, or no
solutions
Definition of Consistent System:
A system is consistent if it has a unique solution or infinitely
many solutions
Definition of Row Echelon Form:
() may have any nonzero value. () may have any values
including 0
Definition of Reduced Row Echelon Form:
() may have any values including 0
The pivot columns are 2, 4, 5, 6, and 9, so the basic variables are
and . The remaining
variables,
and , must be free variables.
Theorem:
If there is no “bad row,” such as 0=1, than the system must be consistent.
Definition of Linear Combination:
Given vector
y
: , is called a linear combination of
v
1
, …,
v
p
with weights
c
1
, …,
c
p
.
Definition of Span :
The set of all linear combinations of .
Example: Let ,
and
Is
w
in the Span ?
Answer: Row reduce the augmented matrix . If there’s no “bad row” then it is in the span
Theorem:
The equation A
x
=
b
has a solution iff
b
is a linear combination of the columns of A.
Definition of Homogeneous Linear Systems:
A
x
=
0
Trivial solution:
Theorem:
The homogeneous equation
has a nontrivial solution iff the equation has at least one
free variable.
Theorem:
A homogeneous system has either a unique solution or infinitely many solutions.
Linear Independence Test:
•
The columns of matrix
are linearly independent iff
the equation
has
only
the trivial
solution
•
A set of two vectors is linearly dependent if one of the vectors is a multiple of the other.
The set is linearly independent iff neither vector is a multiple of the other.
•
An indexed set
of two or more vectors is linearly dependent iff at least one of the vectors
is
S
is a linear combination of the others.
•
A set is linearly dependent if it contains more vectors than there are entries in each vector.
That is, any set
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 Spring '11
 GUEST
 Linear Algebra, Algebra, Linear Equations, Equations, Det, Homogeneous Linear Systems

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