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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 0/trivial subspace

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  • Definition of the 0/trivial subspaceClose match
  • The set containing only 0 is a vector space.Close match
  • SubspacesClose match
  • Definition of subspaceClose match
  • The kernel/null space of a linear transformation is a subspaceClose match
  • Definition of trivial linear dependence relation on a set of vectorsClose match
  • The 0 vector is unique.Close match
  • Definition of 0 vectorClose match
  • A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinationsClose match
  • Definition of kernel/null space of linear transformationClose match
  • If a set of vectors contains the 0 vectorClose match
  • The 0 scalar multiplied by any vector equals the 0 vector.Close match
  • Definition of trivial solution to a homogeneous linear system of equationsClose match
  • Definition of subspace spanned by a set of a set of vectorsClose match
  • The intersection of subspaces is a subspaceClose match
  • Subspaces associated to a linear transformationClose match
  • The inverse image of a subspace under a linear transformation is a subspace.Close match
  • Definition of generalized kernel/null space of linear transformationRelated
  • Definition of sum of subspacesRelated
  • Definition of complement of a subspaceRelated

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