External direct sum

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Given a finite family of vector spaces over the same field [ilmath]F[/ilmath] which we shall write as the tuple [math]\langle(V_i,F)\rangle_{i=1}^n[/math] (which as Mathematicians are lazy we shall write as [ilmath](V_i)^n_{i=1}[/ilmath][note 1]) we define the external direct sum of [ilmath](V_i)_{i=1}^n[/ilmath], which we'll call the vector space [ilmath]V[/ilmath], as follows:

  • [math]V=\mathop{\boxplus}^n_{i=1}V_i=V_1\boxplus V_2\boxplus\cdots\boxplus V_n=[/math][math]\left\{(v_1,v_2,\cdots,v_n)|\ v_i\in V_i\ \forall i\in\{1,2,\cdots,n\}\right\}[/math]
    With [math]v\in V[/math] meaning [math]v=(v_1,v_2,\cdots,v_n)[/math] for some [ilmath]v_i\in V_i\ \forall i[/ilmath]
    We define the operations as follows:
    • Addition:
      Given a [math]u,v\in V[/math] we define [math]u+v=(u_1+v_1,u_2+v_2,\cdots,u_n+v_n)[/math]
    • Scalar multiplication
      Given a [math]v\in V[/math] and a [math]\lambda\in F[/math] we define [math]\lambda v=(\lambda v_1,\lambda v_2,\cdots,\lambda v_n)[/math]


Sometimes the external direct sum is just called the direct sum (and the internal direct sum simply sum) - if you see direct sum you should default to this, the external direct sum (for more see the Addition of vector spaces page)


Some authors use [math]\oplus[/math][1], others use [math]\oplus^\text{ext}[/math][2] for the external direct sum. Authors that use [math]\boxplus[/math] tend to use [math]\oplus[/math] for the internal direct sum!

This leads to the following:

Guide to reading and writing the external direct sum

Usually when writing the sum of vector spaces it is obvious from the context which one you mean. In the following examples we write addition as [ilmath]+[/ilmath] and deduce the type.

  • Let [math]W=U+V[/math]
    to sum them in any way we require that [ilmath]U[/ilmath] and [ilmath]V[/ilmath] be over the same field. It is not mentioned that they are subspaces of a larger space (so any internal sum is out of the question) - we deduce quickly that [ilmath]W[/ilmath] is a vector space over the field of [ilmath]U[/ilmath] and [ilmath]V[/ilmath] and it is defined as: [math]W=U\boxplus V[/math]
    So even if the author had written [math]W=U\oplus V[/math] we could still deduce that the external direct sum was intended.
  • Let [ilmath]W[/ilmath] and [ilmath]U[/ilmath] be subspaces of [ilmath]V[/ilmath], and define [ilmath]S=U+V[/ilmath]
    We're adding subspaces of something, this should scream internal direct sum, also [ilmath]+[/ilmath] is traditionally used to denote such an internal sum (simply called "sum" or "internal direct sum")

See next

TODO: See next section

See also

TODO: More types of vector space sum


  1. See the Tuple page for details - it is okay to use notations [ilmath]\langle\rangle[/ilmath] for tuples when brackets are undesirable and it is easy to see from the context we do not mean the Inner product (which in this case it is because we have a range (the [ilmath]i=1[/ilmath] and [ilmath]n[/ilmath] parts) on the angle brackets, note that [math]\Big((V_i,F)\Big)_{i=1}^n[/math] would be really nasty looking!


  1. Advanced Linear Algebra - Third Edition - Steven Roman - Graduate Texts in Mathematics
  2. According to Advanced Linear Algebra - Third Edition - Steven Roman - Graduate Texts in Mathematics