Linear Algebra, 2020a

Section III. Basis and Dimension 145
also write R 3 = xy-plane + yz-plane. To check this, note that any ⃗w ∈ R 3 can
be written as a linear combination of a member of the xy-plane and a member
of the yz-plane; here are two such combinations.
⎛
⎜
w ⎞ ⎛
1
⎟ ⎜
w ⎞ ⎛ ⎞ ⎛
1 0
⎟ ⎜ ⎟ ⎜
w ⎞ ⎛ ⎞ ⎛ ⎞
1 w 1
0
⎟ ⎜ ⎟ ⎜ ⎟
⎝w 2 ⎠ = 1 · ⎝w 2 ⎠ + 1 · ⎝ 0 ⎠ ⎝w 2 ⎠ = 1 · ⎝w 2 /2⎠ + 1 · ⎝w 2 /2⎠
w 3 0 w 3 w 3 0
The above definition gives one way in which we can think of a space 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 R 3 ,we
often speak of a vector’s ‘x part’ or ‘y part’ or ‘z part’. That is, in our prototype
each vector has a unique decomposition into pieces 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 we
might mean “in such a way that every vector has at least one decomposition,”
which gives 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 Consider R 3 with its standard basis E 3 = 〈⃗e 1 ,⃗e 2 ,⃗e 3 〉. The subspace
with the basis B 1 = 〈⃗e 1 〉 is the x-axis, the subspace with the basis B 2 = 〈⃗e 2 〉 is
the y-axis, and the subspace with the basis B 3 = 〈⃗e 3 〉 is the z-axis. The fact
that any member of R 3 is expressible as a sum of vectors from these subspaces
⎛ ⎞ ⎛
x
⎜ ⎟ ⎜
x
⎞ ⎛ ⎞ ⎛
0
⎟ ⎜ ⎟ ⎜
0
⎞
⎟
⎝y⎠ = ⎝0⎠ + ⎝y⎠ + ⎝0⎠
z 0 0 z
reflects the fact that E 3 spans the space — this equation
⎛ ⎞ ⎛
x
⎜ ⎟ ⎜
1
⎞ ⎛
⎟ ⎜
0
⎞ ⎛
⎟ ⎜
0
⎞
⎟
⎝y⎠ = c 1 ⎝0⎠ + c 2 ⎝1⎠ + c 3 ⎝0⎠
z 0 0 1
w 3