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Definition of intersection of subspaces
Definition of direct sum of subspaces
Definition of the 0/trivial subspace
Definition of subspace
Definition of subspace spanned by a set of a set of vectors
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
The intersection of subspaces is a subspace
Definition of sum of subspaces
The sum of subspaces is a subspace
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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Definition of direct sum of subspacesClose match
The union of bases from independent subspaces is a basis for the space.Close match
Theorem characterizing when a space is the direct sum of two subspacesClose match
Definition of sum of subspacesClose match
Nested finite-dimensional subspaces of equal dimension are equalClose match
The dimension of a subspace is less than or equal to the dimension of the whole spaceClose match
The sum of subspaces is a subspaceClose match
Definition of independent subspacesClose match
DimensionClose match
A vector can be written uniquely as a linear combination of vectors from independent subspaces.Close match
Definition of dimension of a vector space (or subspace)Close match
The direct sum of a subspace and its orthogonal complement is the whole space.Close match
Every basis for a vector space contains the same number of elementsClose match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose match
The intersection of subspaces is a subspaceClose match
Definition of subspace spanned by a set of a set of vectorsClose match
Definition of dimension of a vector space (or subspace) being finite or infiniteClose match
Definition of subspaceRelated
If a vector space has dimension nRelated
Isomorphic vector spaces have the same dimension.Related

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