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  • Coordinate vector spaces
  • Abstract vector spaces

Siblings14
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  • Algebraic properties of R^n (or C^n)
  • Geometric properties of R^n (or C^n)
  • Axioms of a vector space
  • Linear combinations
  • Spans
  • Subspaces
  • Linear (in)dependence
  • Bases
  • Dimension
  • Linear transformations
  • Orthogonality and projection
  • Definition and terminology
  • Basic properties
  • Isomorphism

Children16
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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. Subspaces

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501–525 of 715 nodes
  • Matrix transpose commutes with matrix inverse.#23526
  • Matrix transpose is an involution.#23504
  • Matrix-vector multiplication is a linear transformation.#23856
  • Matrix-vector product is associative#23486
  • Matrix-vector products#23482
  • Multiplication#23487
  • Multiplication by a change of coordinates matrix converts representations for different bases.#23997
  • Multiplication by a Hermitian matrix commutes with the standard inner product on C^n.#23598
  • Multiplication of block/partitioned matrices#23569
  • Multiplicity#23695
  • Multiplying a row by a scalar multiplies the determinant by that scalar.#23739
  • Nilpotent matrices#23580
  • Non-example of a linear transformation#23850
  • Nonsingular matrices and equivalences#23640
  • Normal matrices#23599
  • Norm and length#23907
  • Notation for entry of matrix#23450
  • Notation for the set of m by n matrices#23453
  • Operations on matrices#23460
  • Orthogonality#23914
  • Orthogonality and projection#23893
  • Parametric form of the solution set of a system of linear equations#23419
  • Parametric vector form of the solution set of a system of linear equations#23420
  • Particular types of matrices#23535
  • Projection#23937
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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