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Definition of subspace
Definition of subspace spanned by a set of a set of vectors
Definition of the 0/trivial subspace
Definition of 0/trivial subspace
A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations
Definition of intersection of subspaces
The intersection of subspaces is a subspace
Definition of sum of subspaces
The sum of subspaces is a subspace
Definition of direct sum of subspaces
The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.
Definition of independent subspaces
A vector can be written uniquely as a linear combination of vectors from independent subspaces.
The union of bases from independent subspaces is a basis for the space.
Definition of complement of a subspace
Theorem characterizing when a space is the direct sum of two subspaces
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The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.
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