Dimension of a direct sum

What can be said about the dimension of a direct sum? If is -dimensional, is -dimensional, and , what is the dimension of ? This question is easy to answer.

Theorem 1. The dimension of a direct sum is the sum of the dimensions of its summands.

Proof. We assert that if is a basis in , and if is a basis in , then the set (or, more precisely, the set ) is a basis in . The easiest proof of this assertion is to use the implication (1) (3) from the theorem of the preceding section. Since every in may be written in the form , where is a linear combination of and is a linear combination of , it follows that our set does indeed span . To show that the set is also linearly independent, suppose that The uniqueness of the representation of in the form implies that and hence the linear independence of the ’s and of the ’s implies that ◻

Theorem 2. If is any -dimensional vector space, and if is any -dimensional subspace of , then there exists an -dimensional subspace in such that .

Proof. Let be any basis in ; by the theorem of Section: Bases we may find a set of vectors in with the property that is a basis in . Let be the subspace spanned by ; we omit the verification that . ◻

Theorem 2 says that every subspace of a finite-dimensional vector space has a complement.

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