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Subspaces

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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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A vector can be written uniquely as a linear combination of vectors from independent subspaces.

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The union of bases from independent subspaces is a basis for the space.Close match
Definition of independent subspacesClose match
A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.Close match
Each vector can be written uniquely as a linear combination of vectors from a given basis.Close match
Example of linear combination of vectors in R^2Close match
Linear combinationsClose match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose match
Linear (in)dependenceClose 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
The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.Close match
A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinationsClose match
Definition of sum of subspacesClose match
SubspacesClose match
The sum of subspaces is a subspaceClose match
Theorem characterizing when a space is the direct sum of two subspacesClose match
A set is a basis if each vector can be written uniquely as a linear combination.Close match
Definition of direct sum of subspacesClose match
Definition of subspaceClose match
A set of vectors containing more elements than the dimension of the space must be linearly dependentClose match
The 0 vector is unique.Close match

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