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Kunze Linear Algebra
· Problem 2.1.4
Vector Spaces
Mathematics — Methods · vector-space-axioms
Problem
Let
V
V
V
be the set of all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers, and let
F
F
F
be the field of real numbers. Define
(
x
,
y
)
+
(
x
1
,
y
1
)
=
(
x
+
x
1
,
y
+
y
1
)
(x, y) + (x_1, y_1) = (x + x_1, y + y_1)
(
x
,
y
)
+
(
x
1
,
y
1
)
=
(
x
+
x
1
,
y
+
y
1
)
c
(
x
,
y
)
=
(
c
x
,
y
)
.
c(x, y) = (cx, y).
c
(
x
,
y
)
=
(
c
x
,
y
)
.
Is
V
V
V
, with these operations, a vector space over the field of real numbers?
Related theory
Vector Spaces Definition And Axioms
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