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Coordinate vector spaces

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Algebraic properties of R^n (or C^n)
Axioms of a vector space
Spans
Linear (in)dependence
Bases
Dimension
Linear transformations
Linear combinations
Subspaces
Geometric properties of R^n (or C^n)
Orthogonality and projection

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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
Nested finite-dimensional subspaces of equal dimension are equal
A set of vectors containing more elements than the dimension of the space must be linearly dependent

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Definition of dimension of a vector space (or subspace)Close match
BasesClose match
Definition of dimension of a vector space (or subspace) being finite or infiniteClose match
The rank plus the nullity of a linear transformation equals the dimension of the domain.Close match
If a vector space has dimension nClose match
Every basis for a vector space contains the same number of elementsClose match
The dimension of a subspace is less than or equal to the dimension of the whole spaceClose match
Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.Close match
A set of vectors containing more elements than the dimension of the space must be linearly dependentClose match
SpansClose match
Coordinate vector spacesClose match
IsomorphismClose match
Isomorphic vector spaces have the same dimension.Close match
The set of m by n matrices is a vector space.Close match
Vector spacesClose match
The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.Close match
SubspacesClose match
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.Close match
The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.Close match
Definition of basis of a vector space (or subspace)Close match

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