Sums and direct sums

Sum of subspaces of a vector space

Let \(V\) be a

A vector space over a field \(\mathbb{F}\) is a set \(V\) with two operations:

Addition: \(+:V \times V \rightarrow V\), and
Scalar multiplication: \(\cdot : \mathbb{F} \times V \rightarrow V \)

such that the following axioms hold for all \(u,v,w \in V\) and all \(a,b \in \mathbb{F}\).

Commutativity. \(u+v = v+u\)
Associativity. \( (u+v)+w = u+(v+w) \) and \( (ab)v = a(b v) \)
Additive Identity. Exists \(0 \in V\) such that \(v + 0 =v \)
Additive Inverse. For all \(v \in V\) there exists \(z \in V\) such that \(v+z=0\)
Multiplicative Identity. \(1v=v\)
Distributive Properties. \(a (u + v) = au + av\) and \(a+b)u = au + bu\)

and let \(U, U'\) be

A subset \(U\) of a vector space \(V\) is a subspace of \(V\) if \(U\) is also a vector space over the same field, using the same addition and scalar multiplication rules.

of \(V\).

We define the sum of \(U\) and \(U'\) as the set \( U + U' = \{u + u' : u \in U, u' \in U' \} \) equipped with the addition and scalar multiplication from \(V\).

This sum is a subspace of (V), since

\(0\) is in \(U\) and \(U'\), and by additive identity \(0+0 \in U = U'\)
If \(u,v \in U\) and \(u',v' \in U'\), then \( u+u' \in U + U'\) and \(v + v' \in U + U'\). Adding them \(u+u'+v+v'\) is also in \(U+U'\) by writing in the necessary form using commutativity of addition and associativity: \(u+u'+v+v' = (u+v) + (u'+v')\) which is something in \(U\) plus something in \(U'\).
For scalar multiplication, let \(\alpha\) be a field element. The multiple of an arbitrary element of the sum is \(\alpha(u+u')\). By distributivity, we can write is as something ( \(\alpha u\) ) in \(U\) plus something ( \(\alpha u'\) ) in \(U'\).

This definition extends to a sum of multiple subspaces, or even just subsets of a vector space \(V\).

The sum of subspaces \(U_1, \dots, U_m\) of a vector space \(V\) is the set of all sums of elements, one in each space \( U_1 + \cdots U_m = \{u_1 + \cdots + u_m : u_i \in U_i \} \) equipped with the addition and scalar multiplication from \(V\).

It can also make sense to let the \(U_i\) just be subsets, as in Axler Definition 1.36, but if we do that then the sum defined this way is not guaranteed to be a subspace.

Many mathematical objects have two complementary definitions: one which is constructive (what the object is), and one of which is in terms of properties (what the object does or the axioms it satisfies). The definition of sum given above is the constructive version. As we move up in abstraction, we tend to use more definitions in terms of properties, since then we only need to care about the interface the mathematical object has, rather than the details of how it is made.

Here is the version of the definition of sum in terms of axioms or properties: it is the smallest subspace containing the summands. Since we already have a definition, it is presented as a proposition (1.39 in Axler).

For \(U_i\) subspaces of a vector space \(V\), \(U_1 + \cdots + U_m\) is the smallest subspace of \(V\) containing all the \(U_i\).

In writing proofs, sometimes it is more convenient to use the constructive description of sum (or whatever other piece of mathematics you are studying), and sometimes it is easier to use a more axiomatic or properties-based description. For example, when proving something by contradiction, it often works to use the properties version. In the above case, this might come into play by assuming there is a subspace which is strictly smaller than the sum, but contains all the \(U_i\).

Direct sum of subspaces of a vector spaces

A direct sum is a sum in which the representation of each vector as a sum of elements, one in each summand subspace, is unique (Definition 1.40 in Axler).

A sum of subspaces \(W:=U_1 + \cdots + U_m\) of a vector space \(V\) is a direct sum if every vector \(v\) in the sum has a unique representation as a sum of elements \(v = u_1 + \cdots + u_m\), one in each space. In this case we write the sum as \(W = U_1 \oplus \cdots \oplus U_m\).

The \(\oplus\) tells us that the sum is a direct sum: the decomposition of any vector in the sum into a sum of elements, one in each summand, is unique.

Note that in this definition, there are no scalars modifying the \(u_i\) in the sum. Why not?

It would be inconvenient to check this condition for every vector. So, we prove a Direct Sum Criterion (Propositions 1.44 in Axler) that says it is enough to check that the zero vector has a unique representation.

For \(U_i\) subspaces of a vector space \(V\), \(U_1 + \cdots + U_m\) is a direct sum if whenever \(0 = u_1 + \cdots u_m\), with each \(u_i \in U_i\), we have that each \(u_i =0\).

In the special case of two summands, the criterion is even simpler. If \(U,W\) are subspaces of \(V\), \(U+W\) is a direct sum \( U \cap W \) is the zero vector space.

The word iff is an abbreviation for if and only if..

Here's a video lecture on this material.