Linear Algebra

Section III. Basis and Dimension 137
4.18 Example In R 3 ,thexy-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. A complement of the yz-plane is the line through
(1, 1, 1).
4.19 Example Following Lemma 4.15, here is a natural question: is the simple
sum V = W1 + ···+ Wk also a direct sum if and only if the intersection of the
subspaces is trivial? The answer is that 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 R3 ,let
W1 be the x-axis, let W2 be the y-axis, and let W3 be this.
⎛
W3 = { ⎝ q
⎞
q⎠
r
� � q, r ∈ R}
The check that R3 = W1 + W2 + W3 is easy. The intersection W1 ∩ W2 ∩ W3 is
trivial, but decompositions aren’t unique.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
x 0 0 x x − y 0 y
⎝y⎠
= ⎝0⎠
+ ⎝y
− x⎠
+ ⎝x⎠
= ⎝ 0 ⎠ + ⎝0⎠
+ ⎝y⎠
z 0 0 z 0 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, in this book the direct sum
definition is needed to do the Jordan Form construction in the fifth chapter.
Exercises
� 4.20 Decide if R 2 � �is the direct sum of � each � W1 and W2.
x �� x ��
(a) W1 = { x ∈ R}, W2 = { x ∈ R}
0
x
� � � �
s �� s ��
(b) W1 = { s ∈ R}, W2 = { s ∈ R}
s
1.1s
(c) W1 = R 2 , W2 �= {�0} �
t ��
(d) W1 = W2 = { t ∈ R}
t
� � � � � � � �
1 x �� −1 0 ��
(e) W1 = { + x ∈ R}, W2 = { + y ∈ R}
0 0
0 y
� 4.21 Show that R 3 is the direct sum of the xy-plane with each of these.
(a) the z-axis
(b) the line
� �
z ��
{ z z ∈ R}
z