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If
are nonempty subsets of . The sum of and is the set In general for
, we define the sum as -
A vector space
is said to be the direct sum of and , denoted if such that and In general for subspaces
we have that is the direct sum , denoted if And
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(Friedberg e1.6..20) If
are finite dimensional subspaces such that . Then is also finite dimensional and Moreover,
if and only if -
(Friedberg e1.6.24) If
where is finite-dimensional, then there is a subspace such that - Intuition: A basis of
is linearly independent in . Such a set can be extended to a basis in . The vectors we add to such a basis defines the subspace .
- Intuition: A basis of
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(Friedberg 5.15) Let
where is a finite dimensional vector space. The following are equivalent and for any vectors where - Each vector
can be uniquely written as , . - If for each
, is an ordered basis for then is an ordered basis for. The order comes by first listing all of , then and so on. - For each
., there exists an ordered basis for such that is an ordered basis for . - (Friedberg e5.2.18)
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We can extend direct sums to matrices as well.
Let
, and not necessarily of the same size. The direct sum is the matrix such that In other words
In general if
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(Friedberg 5.30) Let
be a linear operator on a finite-dimensional vector space and let be -invariant subspaces of such that . For each , let be a basis for and . If and . Then