Linear Algebra, 2020a

Section I. Definition of Vector Space 85
multiplication operations are always sensible — they are defined for every pair of
vectors and every scalar and vector, and the result of the operation is a member
of the set.
1.3 Example This subset of R 2 is a line through the origin.
(
L = {
x
y
)
| y = 3x}
We shall verify that it is a vector space under the usual meaning of ‘+’ and ‘·’.
( ) ( ) ( ) ( ) ( )
x 1 x 2 x 1 + x 2
x rx
+ =
r · =
y 1 y 2 y 1 + y 2 y ry
These operations are just the ordinary ones, reused on its subset L. We say that
L inherits these operations from R 2 .
We shall check all ten conditions. The paragraph having to do with addition
has five conditions. For condition (1), closure under addition, suppose that we
start with two vectors from the line L,
⃗v 1 =
( )
x 1
y 1
⃗v 2 =
( )
x 2
y 2
so that they satisfy the restrictions that y 1 = 3x 1 and y 2 = 3x 2 . Their sum
( )
x 1 + x 2
⃗v 1 + ⃗v 2 =
y 1 + y 2
is also a member of the line L because the fact that its second component is
three times its first y 1 + y 2 = 3(x 1 + x 2 ) follows from the restrictions on ⃗v 1
and ⃗v 2 . For (2), that addition of vectors commutes, just compare
⃗v 1 + ⃗v 2 =
( )
x 1 + x 2
y 1 + y 2
⃗v 2 + ⃗v 1 =
( )
x 2 + x 1
y 2 + y 1
and note that they are equal since their entries are real numbers and real numbers
commute. (That the vectors satisfy the restriction of lying in the line is not
relevant for this condition; they commute just because all vectors in the plane
commute.) Condition (3), associativity of vector addition, is similar.
(
(
)
x 1
+
y 1
( )
x 2
)+
y 2
( )
x 3
=
y 3
(
)
(x 1 + x 2 )+x 3
(y 1 + y 2 )+y 3
(
)
x 1 +(x 2 + x 3 )
=
y 1 +(y 2 + y 3 )
( ) ( )
x 1 x 2
= +( +
y 1 y 2
(
x 3
y 3
)
)