Direct sum · aiwiki.page

The direct sum, denoted by $\oplus$ , combines vector spaces or modules into a structure whose elements have uniquely determined components. It can be understood either as a construction of a new space from given spaces, called an external direct sum, or as a decomposition of an existing space into subspaces, called an internal direct sum. For infinitely many components, the algebraic construction requires each element to have only finitely many nonzero components. (math.stanford.edu)

External direct sums

Let $V$ and $W$ be vector spaces over the same field $F$ . Their external direct sum is the set

V\oplus W=\{(v,w):v\in V,\ w\in W\},

with componentwise operations:

(v,w)+(v',w')=(v+v',w+w'),\qquad a(v,w)=(av,aw).

Thus, for two spaces, the underlying set is their Cartesian product. The construction does not require $V$ and $W$ to be subspaces of a common ambient space. Each becomes a subspace of the new space through the canonical inclusions

\iota_V(v)=(v,0),\qquad \iota_W(w)=(0,w).

Every element then has the unique decomposition

(v,w)=\iota_V(v)+\iota_W(w).

The same construction applies to any finite family of vector spaces or modules over a fixed ring. (math.stanford.edu)

Internal direct sums

For subspaces $U,W\subseteq V$ , their ordinary sum is

U+W=\{u+w:u\in U,\ w\in W\}.

This sum is direct when every element has exactly one such expression. Equivalently,

U\cap W=\{0\}.

Writing

V=U\oplus W

asserts both that $U+W=V$ and that the decomposition is unique. Indeed, if $u+w=u'+w'$ , then

u-u'=w'-w\in U\cap W;

a zero intersection therefore forces $u=u'$ and $w=w'$ . The addition map

U\oplus_{\mathrm{external}}W\longrightarrow V,\qquad (u,w)\longmapsto u+w

is then an isomorphism. (linear.pugetsound.edu)

For several subspaces, uniqueness means that

u_1+\cdots+u_n=0,\qquad u_i\in U_i,

implies $u_i=0$ for every $i$ . Pairwise zero intersections alone are insufficient for three or more summands. For example, the three lines in $\mathbb R^2$ spanned by $(1,0)$ , $(0,1)$ , and $(1,1)$ intersect pairwise only at zero, but

(1,0)+(0,1)-(1,1)=0

is a nontrivial relation between them. This illustrates the stronger uniqueness requirement in the definition. (cfm.brown.edu)

Bases, dimension, and complements

If $V=U\oplus W$ , the union of a basis of $U$ and a basis of $W$ is a basis of $V$ . Consequently, in finite dimensions,

\dim V=\dim U+\dim W.

More generally, for finite-dimensional subspaces,

\dim(U+W)=\dim U+\dim W-\dim(U\cap W).

The intersection term measures the overlap that prevents an ordinary sum from being direct. (linear.pugetsound.edu)

A subspace $W$ satisfying $V=U\oplus W$ is called a complement of $U$ . In finite-dimensional spaces, complements can be obtained by extending a basis of $U$ to a basis of $V$ and taking the span of the added vectors. Complements need not be unique. Nor does a direct sum require orthogonality: that additional notion depends on an inner product. In a finite-dimensional inner-product space,

V=U\oplus U^\perp,

where $U^\perp$ is the orthogonal complement of $U$ . (linear.pugetsound.edu)

A decomposition also determines a linear projection:

P(u+w)=u.

It satisfies $P^2=P$ , has image $U$ , and has kernel $W$ . When $W=U^\perp$ , this is the orthogonal projection onto $U$ ; otherwise it is generally an oblique projection. (math.umd.edu)

Infinite algebraic direct sums

For a family $(M_i)_{i\in I}$ of modules over a ring $R$ , the algebraic direct sum is

\bigoplus_{i\in I}M_i = \left\{(m_i)\in\prod_{i\in I}M_i: m_i=0\text{ for all but finitely many }i\right\}.

Addition and scalar multiplication are componentwise. The finite-support condition ensures that each element is a finite sum of elements from the individual components. (math.stanford.edu)

The direct product $\prod_i M_i$ , by contrast, permits unrestricted families of coordinates. The two constructions coincide for a finite index set. Their canonical inclusion is proper when infinitely many $M_i$ are nonzero. For example,

\bigoplus_{n=1}^{\infty}\mathbb R

consists of eventually zero real sequences, whereas $\prod_{n=1}^{\infty}\mathbb R$ contains all real sequences, including $(1,1,1,\ldots)$ . (math.stanford.edu)

Universal property

The direct sum is characterized by a universal property. Given a module $N$ and a family of module homomorphisms

f_i:M_i\longrightarrow N,

there is a unique homomorphism

f:\bigoplus_i M_i\longrightarrow N

such that $f\circ\iota_i=f_i$ for every canonical inclusion $\iota_i$ . Explicitly,

f((m_i))=\sum_i f_i(m_i),

which is well-defined because only finitely many terms are nonzero. In the language of category theory, this makes the algebraic direct sum the coproduct in the category of $R$ -modules. The direct product has the opposite mapping property: maps into it are specified by maps into each component. (math.stanford.edu)

Matrices and representations

For matrices $A$ and $B$ , their direct sum is the block matrix

A\oplus B= \begin{pmatrix} A&0\\ 0&B \end{pmatrix}.

If they represent linear operators on $V$ and $W$ , respectively, this matrix represents the componentwise operator

(A\oplus B)(v,w)=(Av,Bw).

A common block-diagonal form therefore expresses independent action on the summands. (ocw.mit.edu)

In representation theory, two representations $\rho_V$ and $\rho_W$ of the same group combine as

(\rho_V\oplus\rho_W)(g)(v,w) = \bigl(\rho_V(g)v,\rho_W(g)w\bigr).

Decomposing a representation into a direct sum requires complementary invariant subspaces, not merely an arbitrary decomposition of its underlying vector space. Matrix descriptions of such a decomposition use one basis in which every representing matrix has the same block structure. (sites.ualberta.ca)

Hilbert-space direct sums

In functional analysis, “direct sum” can designate a completed construction rather than the finite-support algebraic one. For a sequence of Hilbert spaces $H_n$ , the Hilbert direct sum is

\bigoplus_{n=1}^{\infty}H_n = \left\{(x_n):x_n\in H_n,\ \sum_{n=1}^{\infty}\|x_n\|^2<\infty\right\},

with inner product

\langle x,y\rangle = \sum_{n=1}^{\infty}\langle x_n,y_n\rangle_{H_n}.

It is the completion of the algebraic direct sum in the associated norm. Taking every $H_n=\mathbb R$ or $\mathbb C$ gives the sequence space $\ell^2$ . Infinite square-summable sequences are allowed, unlike in the algebraic direct sum. (web.math.princeton.edu)

A closed subspace $U$ of a Hilbert space has an orthogonal decomposition $H=U\oplus U^\perp$ . The closedness condition matters: algebraic decompositions concern finite sums and uniqueness, whereas Hilbert-space decompositions additionally involve convergence and completeness. (web.math.princeton.edu)

References

Math 210A. Homework 1math.stanford.edu
MATH 210A, FALL 2017: Homework 1 Solutionsmath.stanford.edu
Math 210A: Modern Algebra, Homework 1math.stanford.edu
SCLA Direct Sumslinear.pugetsound.edu
Linear Algebra, Part 3: Direct Sums (Mathematica)cfm.brown.edu
Lecture 12: Direct Sums and Projectionsmath.umd.edu
RES.18-012 (Spring 2022) Full Lecture Notes: Algebra II Student Notesocw.mit.edu
Properties of representationssites.ualberta.ca
Functional Analysis Princeton University MAT520 Lecture Notesweb.math.princeton.edu