Linear Algebra

Section III. Basis and Dimension 133
also write R3 = xy-plane + yz-plane. To check this, we simply note that any
�w ∈ R3 can be written
⎛
⎝ w1
⎞ ⎛
w2⎠
=1· ⎝ w1
⎞ ⎛
w2⎠
+1· ⎝
0
0
⎞
0 ⎠
w3
as a linear combination of a member of the xy-plane and a member of the
yz-plane.
The above definition gives one way in which a space can be thought of as a
combination of some of its parts. However, the prior example shows that there is
at least one interesting property of our benchmark model that is not captured by
the definition of the sum of subspaces. In the familiar decomposition of R3 ,we
often speak of a vector’s ‘x part’or‘y part’or‘z part’. That is, in this model,
each vector has a unique decomposition into parts that come from the parts
making up the whole space. But in the decomposition used in Example 4.4, we
cannot refer to the “xy part” of a vector — these three sums
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 0 1 0 1 0
⎝2⎠
= ⎝2⎠
+ ⎝0⎠
= ⎝0⎠
+ ⎝2⎠
= ⎝1⎠
+ ⎝1⎠
3 0 3 0 3 0 3
all describe the vector as comprised of something from the first plane plus something
from the second plane, but the “xy part” is different in each.
That is, when we consider how R 3 is put together from the three axes “in
some way”, we might mean “in such a way that every vector has at least one
decomposition”, and that leads to the definition above. But if we take it to
mean “in such a way that every vector has one and only one decomposition”
then we need another condition on combinations. To see what this condition
is, recall that vectors are uniquely represented in terms of a basis. We can use
this to break a space into a sum of subspaces such that any vector in the space
breaks uniquely into a sum of members of those subspaces.
4.5 Example The benchmark is R 3 with its standard basis E3 = 〈�e1,�e2,�e3〉.
The subspace with the basis B1 = 〈�e1〉 is the x-axis. The subspace with the
basis B2 = 〈�e2〉 is the y-axis. The subspace with the basis B3 = 〈�e3〉 is the
z-axis. The fact that any member of R 3 is expressible as a sum of vectors from
these subspaces
w3
⎛
⎝ x
⎞ ⎛
y⎠
= ⎝
z
x
⎞ ⎛
0⎠
+ ⎝
0
0
⎞ ⎛
y⎠
+ ⎝
0
0
⎞
0⎠
z
is a reflection of the fact that E3 spans the space — this equation
⎛
⎝ x
⎞ ⎛
y⎠
= c1 ⎝
z
1
⎞ ⎛
0⎠
+ c2 ⎝
0
0
⎞ ⎛
1⎠
+ c3 ⎝
0
0
⎞
0⎠
1