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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 complement of a subspace

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  • Definition of orthogonal complement of a subspaceClose match
  • Definition of subspaceClose match
  • The direct sum of a subspace and its orthogonal complement is the whole space.Close match
  • SubspacesClose match
  • Definition of sum of subspacesClose match
  • The orthogonal complement of a subspace is a subspace.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 direct sum of subspacesClose match
  • A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.Close match
  • The sum of subspaces is a subspaceClose match
  • Theorem characterizing when a space is the direct sum of two subspacesClose match
  • Definition of (not necessarily orthogonal) projection onto a component of a direct sumClose match
  • Definition of intersection of subspacesClose match
  • The union of bases from independent subspaces is a basis for the space.Close match
  • Definition of subspace spanned by a set of a set of vectorsClose match
  • Subspaces associated to a linear transformationClose match
  • The intersection of subspaces is a subspaceClose match
  • The inverse image of a subspace under a linear transformation is a subspace.Related
  • Definition of 0/trivial subspaceRelated
  • Definition of the 0/trivial subspaceRelated

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