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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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  1. Definition of subspace spanned by a set of a set of vectors

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  • The span of a set of vectors is a subspaceClose match
  • Definition of span of a set of vectorsClose match
  • SpansClose match
  • Definition of spanning/generating set for a space or subspaceClose match
  • Definition of subspaceClose match
  • A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinationsClose match
  • SubspacesClose match
  • Definition of sum of subspacesClose match
  • The union of bases from independent subspaces is a basis for the space.Close match
  • A set of vectors containing fewer elements than the dimension of the space cannot spanClose match
  • The image of a spanning set spans the rangeClose match
  • Definition of basis of a vector space (or subspace)Close match
  • Definition of linear combination of vectorsClose match
  • Definition of the 0/trivial subspaceClose match
  • Geometric description of span of a set of vectors in R^n (or C^n)Close match
  • The intersection of subspaces is a subspaceClose match
  • Definition of complement of a subspaceClose match
  • The sum of subspaces is a subspaceClose match
  • A set of nonzero vectors contains (as a subset) a basis for its span.Close match
  • Definition of 0/trivial subspaceClose match

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