Linear algebra MOC

Direct sum vector space

The direct sum of vector spaces is the coproduct of vector spaces. #m/def/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 #m/def/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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