Page 27 - Linear Algebra Done Right

Subspaces
Subspaces
A subset U of V is called a subspace of V if U is also a vector space Some mathematicians 13
(using the same addition and scalar multiplication as on V). For exam- use the term linear
ple, subspace, which means
{(x 1 ,x 2 , 0) : x 1 ,x 2 ∈ F} the same as subspace.
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is a subspace of F .
If U is a subset of V, then to check that U is a subspace of V we
need only check that U satisfies the following:

additive identity
0 ∈ U

closed under addition
u, v ∈ U implies u + v ∈ U;
closed under scalar multiplication
a ∈ F and u ∈ U implies au ∈ U.
The first condition insures that the additive identity of V is in U. The Clearly {0} is the
second condition insures that addition makes sense on U. The third smallest subspace of V
condition insures that scalar multiplication makes sense on U. To show and V itself is the
that U is a vector space, the other parts of the definition of a vector largest subspace of V.
space do not need to be checked because they are automatically satis- The empty set is not a
fied. For example, the associative and commutative properties of addi- subspace of V because
tion automatically hold on U because they hold on the larger space V. a subspace must be a
As another example, if the third condition above holds and u ∈ U, then vector space and a
−u (which equals (−1)u by 1.6) is also in U, and hence every element vector space must
of U has an additive inverse in U. contain at least one
The three conditions above usually enable us to determine quickly element, namely, an
whether a given subset of V is a subspace of V. For example, if b ∈ F, additive identity.
then
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{(x 1 ,x 2 ,x 3 ,x 4 ) ∈ F : x 3 = 5x 4 + b}
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is a subspace of F if and only if b = 0, as you should verify. As another
example, you should verify that

{p ∈P(F) : p(3) = 0}
is a subspace of P(F).
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2
2
The subspaces of R are precisely {0}, R , and all lines in R through
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3
the origin. The subspaces of R are precisely {0}, R , all lines in R 3