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  • Subspaces

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  • Definition of subspace
  • Definition of subspace spanned by a set of a set of vectors
  • Definition of the 0/trivial subspace
  • 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
  • Definition of intersection of subspaces
  • The intersection of subspaces is a subspace
  • Definition of sum of subspaces
  • The sum of subspaces is a subspace
  • Definition of direct sum of subspaces
  • 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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  1. A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations

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