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Nested finite-dimensional subspaces of equal dimension are equal
Definition of dimension of a vector space (or subspace)
If a vector space has dimension n
Every basis for a vector space contains the same number of elements
Definition of dimension of a vector space (or subspace) being finite or infinite
The dimension of a subspace is less than or equal to the dimension of the whole space
A set of vectors containing more elements than the dimension of the space must be linearly dependent

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The dimension of a subspace is less than or equal to the dimension of the whole space

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Nested finite-dimensional subspaces of equal dimension are equalClose match
Every basis for a vector space contains the same number of elementsClose match
Definition of dimension of a vector space (or subspace)Close match
DimensionClose match
The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.Close match
Definition of dimension of a vector space (or subspace) being finite or infiniteClose match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose 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 basis of a vector space (or subspace)Close match
Finite-dimensional vector spaces of equal dimension over the same field are isomorphicClose match
If a vector space has dimension nClose match
Isomorphic vector spaces have the same dimension.Close match
The sum of subspaces is a subspaceClose match
The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.Close match
A set of vectors containing more elements than the dimension of the space must be linearly dependentClose match
Definition of subspaceClose match
Definition of independent subspacesClose match
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.Close match
Definition of subspace spanned by a set of a set of vectorsClose match

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