Linear Algebra
Department of Mathematics
Indian Institute of Technology Guwahati
January – May 2019
MA 102 (
RA, RKS, MGPP, KVK
)
1 / 21
Vector spaces
Topics:
Vector Spaces and Subspaces
Linear Independence
Basis and Dimension
2 / 21
Field axioms
A
field
is a set
F
with two binary operations called
addition,
denoted by
+,
and
multiplication,
denoted by
·
,
satisfying the
following
field axioms:
3 / 21
Field axioms
A
field
is a set
F
with two binary operations called
addition,
denoted by
+,
and
multiplication,
denoted by
·
,
satisfying the
following
field axioms:
1
Closure:
For all
α, β
∈
F
, the sum
α
+
β
∈
F
and the product
α
·
β
∈
F
.
3 / 21
Field axioms
A
field
is a set
F
with two binary operations called
addition,
denoted by
+,
and
multiplication,
denoted by
·
,
satisfying the
following
field axioms:
1
Closure:
For all
α, β
∈
F
, the sum
α
+
β
∈
F
and the product
α
·
β
∈
F
.
2
Commutativity:
For all
α, β
∈
F
, α
+
β
=
β
+
α
and
α
·
β
=
β
·
α
.
3 / 21
Field axioms
A
field
is a set
F
with two binary operations called
addition,
denoted by
+,
and
multiplication,
denoted by
·
,
satisfying the
following
field axioms:
1
Closure:
For all
α, β
∈
F
, the sum
α
+
β
∈
F
and the product
α
·
β
∈
F
.
2
Commutativity:
For all
α, β
∈
F
, α
+
β
=
β
+
α
and
α
·
β
=
β
·
α
.
3
Associativity:
For all
α, β, γ,
(
α
+
β
) +
γ
=
α
+ (
β
+
γ
) and
(
α
·
β
)
·
γ
=
α
·
(
β
·
γ
).
3 / 21
Field axioms
A
field
is a set
F
with two binary operations called
addition,
denoted by
+,
and
multiplication,
denoted by
·
,
satisfying the
following
field axioms:
1
Closure:
For all
α, β
∈
F
, the sum
α
+
β
∈
F
and the product
α
·
β
∈
F
.
2
Commutativity:
For all
α, β
∈
F
, α
+
β
=
β
+
α
and
α
·
β
=
β
·
α
.
3
Associativity:
For all
α, β, γ,
(
α
+
β
) +
γ
=
α
+ (
β
+
γ
) and
(
α
·
β
)
·
γ
=
α
·
(
β
·
γ
).
4
Identity:
There exist 0
∈
F
and 1
∈
F
such that
α
+ 0 =
α
and 1
·
α
=
α
for all
α
∈
F
.
3 / 21
Field axioms
A
field
is a set
F
with two binary operations called
addition,
denoted by
+,
and
multiplication,
denoted by
·
,
satisfying the
following
field axioms:
1
Closure:
For all
α, β
∈
F
, the sum
α
+
β
∈
F
and the product
α
·
β
∈
F
.
2
Commutativity:
For all
α, β
∈
F
, α
+
β
=
β
+
α
and
α
·
β
=
β
·
α
.
3
Associativity:
For all
α, β, γ,
(
α
+
β
) +
γ
=
α
+ (
β
+
γ
) and
(
α
·
β
)
·
γ
=
α
·
(
β
·
γ
).
4
Identity:
There exist 0
∈
F
and 1
∈
F
such that
α
+ 0 =
α
and 1
·
α
=
α
for all
α
∈
F
.
5
Inverse:
For
α
∈
F
, there exist
β, γ
∈
F
such that
α
+
β
= 0,
and
α
·
γ
= 1 when
α
6
= 0
.
3 / 21
Field axioms
A
field
is a set
F
with two binary operations called
addition,
denoted by
+,
and
multiplication,
denoted by
·
,
satisfying the
following
field axioms:
1
Closure:
For all
α, β
∈
F
, the sum
α
+
β
∈
F
and the product
α
·
β
∈
F
.
2
Commutativity:
For all
α, β
∈
F
, α
+
β
=
β
+
α
and
α
·
β
=
β
·
α
.
3
Associativity:
For all
α, β, γ,
(
α
+
β
) +
γ
=
α
+ (
β
+
γ
) and
(
α
·
β
)
·
γ
=
α
·
(
β
·
γ
).
4
Identity:
There exist 0
∈
F
and 1
∈
F
such that
α
+ 0 =
α
and 1
·
α
=
α
for all
α
∈
F
.
5
Inverse:
For
α
∈
F
, there exist
β, γ
∈
F
such that
α
+
β
= 0,
and
α
·
γ
= 1 when
α
6
= 0
. β
is denoted by

α
and
γ
by
α

1
or 1
/α.
3 / 21
Fields axioms (cont.)