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  • Dimension

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  • Definition of dimension of a vector space (or subspace)
  • If a vector space has dimension n
  • Every basis for a vector space contains the same number of elements
  • Definition of dimension of a vector space (or subspace) being finite or infinite
  • The dimension of a subspace is less than or equal to the dimension of the whole space
  • If two finite dimensional subspaces have the same dimension and one is contained in the other
  • A set of vectors containing more elements than the dimension of the space must be linearly dependent
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  1. A set of vectors containing more elements than the dimension of the space must be linearly dependent

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701–715 of 715 nodes
  • Triangular matrices#23561
  • Two matrices of the same size are equivalent if and only if they have the same rank.#23607
  • Two vectors are orthogonal if and only if the Pythagorean Theorem holds.#23918
  • Unitary matrices#23587
  • Unitary matrices are invertible.#23589
  • Unitary matrices have orthogonal (orthonormal) rows/columns.#23590
  • Unitary matrices preserve inner products.#23591
  • Unitary matrices preserve orthogonal (orthonormal) bases.#23592
  • Using matrices to solve linear systems#23760
  • Vector space isomorphism is an equivalence relation.#24003
  • Vector spaces#23775
  • Vector spaces with the same dimension are isomprphic.#24005
  • Vector sum/addition interpreted geometrically in R^n (or C^n)#23797
  • Vector sum/addition is commutative and associative#23787
  • Visualise a linear transformation on R^2 by looking at the image of a region#23852
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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