Linear algebra MOC

Direct sum vector space

The direct sum of vector spaces is the coproduct of vector spaces. linalg It may be constructed as tuples with componentwise operations (cf. Direct sum of modules).

Internal direct sum

Let 𝑉 be a vector space and {𝑆𝑖}𝑖∈𝐼 be a family of subspaces. Then 𝑉 is the direct sum ⨁𝑖∈𝐼𝑆𝑖 iff 𝑉 =∑𝑖∈𝐼𝑆𝑖 and linalg

𝑆𝑖∩⎛⎜ ⎜⎝∑𝑗≠𝑖𝑆𝑖⎞⎟ ⎟⎠={0}

If 𝑆1⨁𝑆2 =𝑉, then 𝑆2 is a complement of 𝑆1.¹

Further characterisations

Fixed basis

Let 𝑉,𝑊 ∈𝖵𝖾𝖼𝗍𝕂 be vector spaces over 𝕂 with bases {𝑣𝑖}𝑛𝑖=1 and {𝑤𝑗}𝑚𝑗=1 respectively. The direct sum 𝑉 ⊕𝑊 of these spaces then has basis {𝑣𝑖}𝑛𝑖=1 ⨿{𝑤𝑗}𝑚𝑗=1.

Inner product spaces

If 𝑉 and 𝑊 are inner product spaces, then ⟨(𝑣1,𝑤1)|(𝑣2,𝑤2)⟩ =⟨𝑣1|𝑣2⟩ +⟨𝑤1|𝑤2⟩

Properties

• dim⁡(𝑉 ⊕𝑊) =dim⁡𝑉 +dim⁡𝑊

See also

• Direct sum of linear maps
• Direct sum of representations

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Footnotes

1. 2008. Advanced Linear Algebra, pp. 41–42 ↩