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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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A set of vectors containing more elements than the dimension of the space must be linearly dependent

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If a set of vectors in R^n (or C^n) contains more than n elementsClose match
If a vector space has dimension nClose match
Linear (in)dependenceClose match
Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectorsClose match
A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.Close match
Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectorsClose match
DimensionClose match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose match
Definition of linearly independent set of vectors: if a linear combination is 0Close match
A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.Close match
A set of vectors containing fewer elements than the dimension of the space cannot spanClose match
Two vectors are dependent iff at least one is a scalar multiple of the otherClose match
Definition of linear dependence relation on a set of vectorsClose match
Definition of dimension of a vector space (or subspace)Close match
Definition of dimension of a vector space (or subspace) being finite or infiniteClose match
Every basis for a vector space contains the same number of elementsClose match
If a set of vectors contains the 0 vectorClose match
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0Close match
A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.Close match
Definition of trivial linear dependence relation on a set of vectorsClose match

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