Linear Algebra, 2020a

Section I. Definition of Vector Space 97
and scalar multiplication is also the same as in R 3 . To show that P is a
subspace we need only note that it is a subset and then verify that it is a
space. We won’t check all ten conditions, just the two closure ones. For closure
under addition, note that if the summands satisfy that x 1 + y 1 + z 1 = 0 and
x 2 + y 2 + z 2 = 0 then the sum satisfies that (x 1 + x 2 )+(y 1 + y 2 )+(z 1 + z 2 )=
(x 1 + y 1 + z 1 )+(x 2 + y 2 + z 2 )=0. For closure under scalar multiplication, if
x + y + z = 0 then the scalar multiple has rx + ry + rz = r(x + y + z) =0.
2.3 Example The x-axis in R 2 is a subspace, where the addition and scalar
multiplication operations are the inherited ones.
( ) ( ) ( ) ( ) ( )
x 1 x 2 x 1 + x 2 x rx
+ =
r · =
0 0 0
0 0
As in the prior example, to verify directly from the definition that this is a
subspace we simply note that it is a subset and then check that it satisfies the
conditions in definition of a vector space. For instance the two closure conditions
are satisfied: adding two vectors with a second component of zero results in a
vector with a second component of zero and multiplying a scalar times a vector
with a second component of zero results in a vector with a second component of
zero.
2.4 Example Another subspace of R 2 is its trivial subspace.
( )
0
{ }
0
Any vector space has a trivial subspace {⃗0 }. At the opposite extreme, any
vector space has itself for a subspace. A subspace that is not the entire space is
a proper subspace.
2.5 Example Vector spaces that are not R n ’s also have subspaces. The space of
cubic polynomials {a + bx + cx 2 + dx 3 | a, b, c, d ∈ R} has a subspace comprised
of all linear polynomials {m + nx | m, n ∈ R}.
2.6 Example Another example of a subspace that is not a subset of an R n followed
the definition of a vector space. The space in Example 1.12 of all real-valued
functions of one real variable {f | f: R → R} has the subspace in Example 1.14
of functions satisfying the restriction (d 2 f/dx 2 )+f = 0.
2.7 Example The definition requires that the addition and scalar multiplication
operations must be the ones inherited from the larger space. The set S = {1} is
a subset of R 1 . And, under the operations 1 + 1 = 1 and r · 1 = 1 the set S is
a vector space, specifically, a trivial space. However, S is not a subspace of R 1
because those aren’t the inherited operations, since of course R 1 has 1 + 1 = 2.