Linear Algebra, 2020a

Section III. Basis and Dimension 149
4.16 Example In R 2 the x-axis and the y-axis are complements, that is, R 2 =
x-axis ⊕ y-axis. This points out that subspace complement is slightly different
than set complement; the x and y axes are not set complements because their
intersection is not the empty set.
A space can have more than one pair of complementary subspaces; another
pair for R 2 are the subspaces consisting of the lines y = x and y = 2x.
4.17 Example In the space F = {a cos θ + b sin θ | a, b ∈ R}, the subspaces W 1 =
{a cos θ | a ∈ R} and W 2 = {b sin θ | b ∈ R} are complements. The prior example
noted that a space can be decomposed into more than one pair of complements.
In addition note that F can has more than one pair of complementary
subspaces where the first in the pair is W 1 — another complement of W 1 is
W 3 = {b sin θ + b cos θ | b ∈ R}.
4.18 Example In R 3 , the xy-plane and the yz-planes are not complements, which
is the point of the discussion following Example 4.4. One complement of the
xy-plane is the z-axis.
Here is a natural question that arises from Lemma 4.15: for k>2is the
simple sum V = W 1 + ···+ W k also a direct sum if and only if the intersection
of the subspaces is trivial?
4.19 Example If there are more than two subspaces then having a trivial intersection
is not enough to guarantee unique decomposition (i.e., is not enough to
ensure that the spaces are independent). In R 3 ,letW 1 be the x-axis, let W 2 be
the y-axis, and let W 3 be this.
⎛
⎜
q
⎞
⎟
W 3 = { ⎝q⎠ | q, r ∈ R}
r
The check that R 3 = W 1 + W 2 + W 3 is easy. The intersection W 1 ∩ W 2 ∩ W 3 is
trivial, but decompositions aren’t unique.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
x
⎜ ⎟ ⎜
0 0
⎟ ⎜ ⎟ ⎜
x ⎟
⎝y⎠ = ⎝0⎠ + ⎝y − x⎠ + ⎝x⎠ =
z 0 0 z
⎜
⎝
x − y
0
0
⎟
⎠ +
⎜
0 ⎟ ⎜
y ⎟
⎝0⎠ + ⎝y⎠
0 z
(This example also shows that this requirement is also not enough: that all
pairwise intersections of the subspaces be trivial. See Exercise 30.)
In this subsection we have seen two ways to regard a space as built up from
component parts. Both are useful; in particular we will use the direct sum
definition at the end of the Chapter Five.