Download Unit 3: Linear Systems, Vector Spaces

Unit 3: Linear Systems, Vector Spaces
Definition. [–] A set V is called a linear space or a vector space over R if
there are two operations + and · with the following properties
• v 1 , v2 ∈ V ⇒ v 1 + v 2 ∈ V
• v 1 + v 2 = v2 + v 1
• v1 + (v2 + v3) = (v1 + v2) + v3
• There is an element 0V ∈ V with 0V + v = v + 0V = v
• There is an element −v ∈ V with v + (−v) = 0V
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3.1: Vector Spaces (Cont.)
• α∈R⇒α·v ∈V
• (α + β) · v = α · v + β · v
• α · (β · u) = (αβ) · u
• α · (v1 + v2) = α · v1 + α · v2
• 1·v =v
One can then easily show 0 · v = 0 and −1 · v = −v.
The elements v of V are called vectors, the elements of R scalars.
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3.2: Example of Vector Spaces
In linear Algebra: R, the nullspace and the column space of a matrix
In this course:
• space of all n × m matrices
• space of all polynomials of degree n: P n
• space of all continuous functions C[a, b]
• space of all functions with a continuous first derivative C 1[a, b]
• ....
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3.3: Example of Vector Spaces
Not a vector space: The set of all polynomials of degree n which have the
property p(2) = 5.
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3.4: Basis, Coordinates
One describes a certain element in a vector space by giving its coordinates in
a given basis. (Recall these corresponding definitions from linear algebra).
The number of basis vectors determines the dimension of the vector space.
C[a, b] is a vector space with an infinite dimension, P n has finite dimension.
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3.5: Norms
First we recall properties of the absolute value (absolut belopp) of a real
number:
�
v if v ≥ 0
v ∈ R �→ |v| =
−v if v < 0
• |v| ≥ 0 and |v| = 0 ⇔ v = 0
• |λv| = |λ||v|
• |u + v| ≤ |u| + |v|
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3.6: Vector Norms
We generalize the definition of the absolute value of a real number to norms
of vectors and matrices (later also of functions):
Definition. [2.6] V a linear space , a function � · � : V → R is called a
norm if for all u, v ∈ V and all λ ∈ R:
• �v� ≥ 0 and �v� = 0 ⇔ v = 0 (Positivity)
• �λv� = |λ|�v� (Homogenity)
• �u + v� ≤ �u� + �v� (Triangular inequality)
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3.7: Examples
Examples for norms in Rn:
• 1-norm
�v�1 =
n
�
i=1
|vi|
• 2-norm (Euclidean norm)
�v�2 =
• ∞-norm
n
��
i=1
�
2 1/2
vi
max |vi|
i=1:n
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3.8: Unit Circle
The unit circle is the set of all vectors of norm 1.
inf
1
1
0
2
-1
-1
0
1
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3.9: Convergence
Theorem. [-]
If dim V < ∞ and if � · �p and � · �q are norms in V, then there exist
constants c, C such that for all v ∈ V
c�v�q ≤ �v�p ≤ C�v�q
Norms in finite dimensional spaces are equivalent.
Sequences convergent in one norm are convergent in all others.
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3.10: Matrix norms
A matrix defines a linear map.
Definition. [2.10]
Let � · � be a given vector norm. The corresponding (subordinate) matrix
norm is defined as
�Av�
�A� = max
v∈Rn \{0} �v�
I.e. the largest relative change of a vector, when mapped by A.
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3.11: Matrix norms
Example: A =
�
�
2 1
1 1
2
A
0
-2
-1 0 1
-1 0 1
�A� = 2.6180
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3.12: How to compute matrix norms
Matrix norms are computed by applying the following formulas:
1-norm (Th. 2.8): �A�1 = maxj=1:n
�n
i=1 |aij |
∞-norm (Th. 2.7): �A�1 = maxi=1:n
2-norm (Th. 2.9): �A�2 = maxi=1:n
�n
�
j=1 |aij |
maximal column sum
maximal row sum
λi(ATA)
where λi(ATA) is the ith eigenvalue of ATA.
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