Linear Algebra, 2020a

144 Chapter Two. Vector Spaces
III.4
Combining Subspaces
This subsection is optional. It is required only for the last sections of
Chapter Three and Chapter Five and for occasional exercises. You can
pass it over without loss of continuity.
One way to understand something is to see how to build it from component
parts. For instance, we sometimes think of R 3 put together from the x-axis,
the y-axis, and z-axis. In this subsection 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 prototype.
Subspaces are subsets and sets combine via union. But taking the combination
operation for subspaces to be the simple set union operation isn’t what we want.
For instance, the union of the x-axis, the y-axis, and z-axis is not all of R 3 .In
fact this union is not a subspace because it is not closed under addition: this
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. Therefore to combine
subspaces, in addition to the members of those subspaces, we must at least also
include all of their linear combinations.
4.1 Definition Where W 1 ,...,W k are subspaces of a vector space, their sum is
the span of their union W 1 + W 2 + ···+ W k =[W 1 ∪ W 2 ∪···W k ].
Writing ‘+’ fits with the conventional practice of using this symbol for a natural
accumulation operation.
4.2 Example Our R 3 prototype works with this. Any vector ⃗w ∈ R 3 is a linear
combination c 1 ⃗v 1 + c 2 ⃗v 2 + c 3 ⃗v 3 where ⃗v 1 is a member of the x-axis, etc., in this
way
⎛
⎜
w ⎞ ⎛
1
⎟ ⎜
⎝w 2 ⎠ = 1 · ⎝
w 3
w 1
0
0
⎞
⎟
⎠ + 1 ·
⎛ ⎞ ⎛ ⎞
0 0
⎜ ⎟ ⎜ ⎟
⎝w 2 ⎠ + 1 · ⎝ 0 ⎠
0 w 3
and so x-axis + y-axis + z-axis = R 3 .
4.3 Example A sum of subspaces can be less than the entire space. Inside of P 4 ,
let L 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 P 4 .
Instead, 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, wecan