Results from Linear Algebra
We first state the basic existence/uniqueness theorem for 1storder linear systems and examine some
of the consequences.
Theorem
(Existence/Uniqueness) The 1storder linear homogeneous IVP
˙
y
=
A
y
,
y
(
x
0
) =
y
0
always has a unique solution, which is defined for every
x
in the inverval
(
∞
,
∞
)
.
Remark:
In the above theorem, by
y
0
we mean the vector of initial values of the functions
y
1
, y
2
, . . . , y
n
, that is to say the vector
y
0
=
y
0
1
y
0
2
.
.
.
y
0
n
.
Recall the following definitions from linear algebra:
Definition
A
vector space
is a set of objects that is closed under the operations of addition and scalar multi
plication (and a distributive property).
A set of vectors
v
1
, v
2
, . . . , v
n
are
linearlyindependent
if whenever
c
1
v
1
+
c
2
v
2
+
· · ·
+
c
n
v
n
= 0
for some scalars
c
1
, c
2
, . . . , c
n
, then necessarily all of the
c
i
must be 0.
The vectors
v
1
, v
2
, . . . , v
n
form a
basis
for the vector space
V
if they are linearlyindependent and
every vector
v
in
V
can be written in the form
v
=
c
1
v
1
+
c
2
v
2
+
· · ·
+
c
n
v
n
for some scalars
c
1
, c
2
, . . . , c
n
.
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 '08
 staff
 Linear Algebra, Algebra, Linear Systems, Vector Space, vector space Rn

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