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Mathematics 1
Applied Informatics
Štefan BEREŽNÝ
6th lecture
Contents
LINEAR ALGEBRA
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Vector Space
n-dimensional Arithmetic Vector Space
Basis
Dimension
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Vector Space
Definition:
The nonempty set V with the operations “addition”
and “multiplication by real numbers”, which satisfy
following conditions:
(1)
 u, v  V:
u+vV
(i. e. V is closed with respect to operation “addition”)
(2)
 u  V,    R
u  V
(i. e. V is closed with respect to operation “multiplication”)
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Vector Space
(3)
(4)
(5)
(6)
(7)
(8)
 u, v  V:
u+v=v+u
 u, v, w  V:(u + v) + w = v + (u + w)
 u  V:
1u = u
 u  V,  ,   R:
(u) = ()u
 u, v  V,    R:
(u + v) = u + v
 u  V,  ,   R:
( + )u = u + u
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Vector Space
(9)
There exists a so called zero vector
o = (0, 0, …, 0) in V.
If u is any vector from V then:
u+o=u
(10) To every vector u  V there exists
a vector u (vector u a so called opposite
vector to vector u so that:
u + (u) = o
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Vector Space
is called vector space. The elements are called
vectors.
If V = Rn (Rn is set of all n-tuples of real numbers)
then the set V is called the n-dimensional
arithmetic space.
Its elements (n-tuples of real numbers) are called
arithmetic vectors.
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Vector Space
Theorem: (Uniqueness of the zero vector)
Let V is the vector space. There exists only one
zero vector in the vector space V.
Theorem: Let V is the vector space. For any
vector u  V and any real number   R, it holds:
(a) 0u = o
(b) (1)u = u
(c)
o = o
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Vector Space
Definition:
If u1, u2, … , un is a group of vectors in the vector
space V and 1, 2, …, n are real numbers then
the vector v  V:
v = 1u1 + 2u2 + 3u3 + … + nun
is called the linear combination of the vectors
u 1, u 2, … , u n.
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Vector Space
Definition:
If all coefficient 1, 2, …, n are zero then it is
called trivial linear combination.
If there exists coefficient 1, 2, …, n such that at
least one of them is different from zero then it is
called non-trivial linear combination.
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Vector Space
Definition:
The group of vectors u1, u2, … , un is called
linearly dependent if there exists non-trivial linear
combination such that
1u1 + 2u2 + 3u3 + … + nun = o.
A group of vectors u1, u2, … , un which is not
linearly dependent is called linearly independent.
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Vector Space
Theorem:
If one of the vectors u1, u2, … , un in the vector
space V is equal to the zero vector then the group
of vectors u1, u2, … , un is linearly dependent.
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Vector Space
Theorem:
The group of vectors u1, u2, … , un from the vector
space V, where n  1, is linearly dependent if and
only if at least one vector from this group can be
expressed as a linear combination of the other
vectors of the group.
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Vector Space
Theorem:
The group of vectors u1, u2, … , un from the vector
space V is linearly independent if and only if the
vector equation
1u1 + 2u2 + 3u3 + … + nun = o
has only the zero (trivial) solution
1 = 0, 2 = 0, 3 = 0, …, n = 0.
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Vector Space
Theorem:
Let u1, u2, … , un be a linearly dependent group of
vectors from the vector space V. Then every group
of vectors from vector space V which contains the
vectors u1, u2, … , un is also linearly dependent.
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Vector Space
Definition:
Let the group of vectors u1, u2, … , un is from the
vector space V. We say that the set of vectors B =
=u1, u2, … , un is basis of the vector space V if:
(a)
the group of vectors u1, u2, … , un is linearly
independent
(b)
the group of vectors u1, u2, … , un generate
the vector space V.
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Vector Space
Definition:
Let n be a natural number. We say that the vector
space V is n-dimensional or equivalently:
its dimension is equal to n (we write dim(V) = n)
if cardinality of basis of the vector space V is equal
n.
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Vector Space
Theorem:
Let u1, u2, … , un be a basis of the vector space V.
Then every vector from V can be uniquely
expressed as a linear combination of the vectors
u1, … , un. If the vector space V is n-dimensional
then there exists a group of n vectors in V (basis)
which is linearly independent and each group of
more then n vectors from V is linearly dependent.
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Vector Space
Theorem:
Suppose that W is a subset of vector space V.
If W is the vector space with the same operations
“addition” and “multiplication by real numbers” as
in vector space V then we call the set W the
subspace of the vector space V.
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Thank you for your attention.
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