Linear Algebra

132 Chapter 2. Vector Spaces
by finishing the analysis, in the sense that ‘analysis’ means “method of determining
the ... essential features of something by separating it into parts”
[Macmillan Dictionary].
A common way to understand things is to see how they can be built from
component parts. For instance, we think of R 3 as put together, in some way,
from the x-axis, the y-axis, and z-axis. In this subsection we will make this
precise; we will describe how to decompose a vector space into a combination of
some of its subspaces. In developing this idea of subspace combination, we will
keep the R 3 example in mind as a benchmark model.
Subspaces are subsets and sets combine via union. But taking the combination
operation for subspaces to be the simple union operation isn’t what we
want. For one thing, the union of the x-axis, the y-axis, and z-axis is not all of
R 3 , so the benchmark model would be left out. Besides, union is all wrong for
this reason: a union of subspaces need not be a subspace (it need not be closed;
for instance, this R3 vector
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 0 1
⎝0⎠
+ ⎝1⎠
+ ⎝0⎠
= ⎝1⎠
0 0 1 1
is in none of the three axes and hence is not in the union). In addition to the
members of the subspaces, we must at a minimum also include all possible linear
combinations.
4.1 Definition Where W1,...,Wk are subspaces of a vector space, their sum
is the span of their union W1 + W2 + ···+ Wk =[W1 ∪ W2 ∪ ...Wk].
(The notation, writing the ‘+’ between sets in addition to using it between
vectors, fits with the practice of using this symbol for any natural accumulation
operation.)
4.2 Example The R 3 model fits with this operation. Any vector �w ∈ R 3 can
be written as a linear combination c1�v1 + c2�v2 + c3�v3 where �v1 is a member of
the x-axis, etc., in this way
⎛
⎝ w1
⎞ ⎛
w2⎠
=1· ⎝ w1
⎞ ⎛
0 ⎠ +1· ⎝
0
0
⎞ ⎛
w2⎠
+1· ⎝
0
0
0
w3
and so R 3 = x-axis + y-axis + z-axis.
4.3 Example A sum of subspaces can be less than the entire space. Inside of
P4, letL be the subspace of linear polynomials {a + bx � � a, b ∈ R} and let C be
the subspace of purely-cubic polynomials {cx 3 � � c ∈ R}. Then L + C is not all
of P4. Instead, it is the subspace L + C = {a + bx + cx 3 � � a, b, c ∈ R}.
4.4 Example A space can be described as a combination of subspaces in more
than one way. Besides the decomposition R 3 = x-axis + y-axis + z-axis, we can
w3
⎞
⎠