Linear Algebra, 2020a

Section I. Definition of Vector Space 87
1.4 Example The whole plane, the set R 2 , is a vector space where the operations
‘+’ and ‘·’ have their usual meaning.
( ) ( ) ( ) ( ) ( )
x 1 x 2 x 1 + x 2
x rx
+ =
r · =
y 1 y 2 y 1 + y 2 y ry
We shall check just two of the conditions, the closure conditions.
For (1) observe that the result of the vector sum
( ) ( ) ( )
x 1 x 2 x 1 + x 2
+ =
y 1 y 2 y 1 + y 2
is a column array with two real entries, and so is a member of the plane R 2 .In
contrast with the prior example, here there is no restriction on the first and
second components of the vectors.
Condition (6) is similar. The vector
( ) ( )
x rx
r · =
y ry
has two real entries, and so is a member of R 2 .
In a similar way, each R n is a vector space with the usual operations of vector
addition and scalar multiplication. (In R 1 , we usually do not write the members
as column vectors, i.e., we usually do not write ‘(π)’. Instead we just write ‘π’.)
1.5 Example Example 1.3 gives a subset of R 2 that is a vector space. For contrast,
consider the set of two-tall columns with entries that are integers, under the
same operations of component-wise addition and scalar multiplication. This
is a subset of R 2 but it is not a vector space: it is not closed under scalar
multiplication, that is, it does not satisfy condition (6). For instance, on the left
below is a vector with integer entries, and a scalar.
( ) ( )
4 2
0.5 · =
3 1.5
On the right is a column vector that is not a member of the set, since its entries
are not all integers.
1.6 Example The one-element set
⎛ ⎞
0
0
{ ⎜ ⎟
⎝0⎠ }
0