Recall that we know how to add two signal/vectors and scale a
vector/signal.
A vector space
VV is a non empty subset of vectors
such that given xx,
y∈V
y
V
,
then
x+y∈V
x
y
V
, and given
x∈V
x
V
,
and a scalar αα, then
αx∈V
α
x
V
.
In other words, a vector space is closed under
addition and scaling.
Depending on whether
α∈ℝ
α
or ℂ we can distinguish real
vectors spaces from complex vector spaces.
ℝ,
ℝ2
2
,
ℝ3
3
,
......
ℝN
N
ℂ,
ℂ2
2
,
ℂ3
3
,
......
ℂN
N
a line in
ℝ2
2
through the origin
a line in
ℝ3
3
through the origin
a plane in
ℝ3
3
through the origin
...
0
0
A subspace of a vector space is a
subset of the vectors that is itself a vector space.
Vectors lying in a line through the origin in
ℝ2
2
:
Fundamental idea in DSP: split the vector space of all
signals into "signal space" and "noise space".
Why is this reasonable?
Show that the set of all linear combinations of
MM vectors in
ℝN
N
is a vector space.
If a vector space consists of all linear combinations of
MM vectors, then these vectors
span the vector space. In other words, the
spanning vectors can generate a vector in the vector space.
The vector
11
1
1
spans this subspace in
ℝ2
2
:
The canonical δδ basis,
δ0...δN-1
δ
0
...
δ
N
1
,
span
ℝN
N
.
Can
M<N
M
N
vectors span
ℝN
N
?
Do
M≥N
M
N
vectors always span
ℝN
N
?
Is
ℝN
N
a subspace if
ℂN
N
?