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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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If a vector space has dimension n

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A set of vectors containing more elements than the dimension of the space must be linearly dependentClose match
Every basis for a vector space contains the same number of elementsClose match
Definition of basis of a vector space (or subspace)Close match
DimensionClose match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose match
Definition of dimension of a vector space (or subspace)Close match
A set of vectors containing fewer elements than the dimension of the space cannot spanClose match
A set of nonzero vectors contains (as a subset) a basis for its span.Close match
BasesClose match
A set is a basis if each vector can be written uniquely as a linear combination.Close match
Definition of dimension of a vector space (or subspace) being finite or infiniteClose match
The standard/natural basis of R^n (or C^n) is a basis.Close match
Isomorphic vector spaces have the same dimension.Close match
Geometric description of span of a set of vectors in R^n (or C^n)Close match
SpansClose match
If a set of vectors in R^n (or C^n) contains more than n elementsClose match
Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.Close match
If a set of vectors contains the 0 vectorClose match
The dimension of a subspace is less than or equal to the dimension of the whole spaceClose match
Nested finite-dimensional subspaces of equal dimension are equalClose match

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