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 be a vector space and be a family of subspaces. Then is the direct sum iff and linalg

If , then is a complement of .¹

Further characterisations

Fixed basis

Let be vector spaces over with bases and respectively. The direct sum of these spaces then has basis .

Inner product spaces

If and are inner product spaces, then

Properties

See also

• Direct sum of linear maps
• Direct sum of representations

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Footnotes

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