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  • Definition of basis of a vector space (or subspace)
  • Definition of the standard/natural basis of R^n (or C^n)
  • The standard/natural basis of R^n (or C^n) is a basis.
  • Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)
  • Definition of the standard basis of the polynomials of degree at most n
  • Definition of the standard basis of the m by n matrices
  • Definition of coordinates relative to a given basis
  • A set of nonzero vectors contains (as a subset) a basis for its span.
  • The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.
  • Each vector can be written uniquely as a linear combination of vectors from a given basis.
  • A set is a basis if each vector can be written uniquely as a linear combination.
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  1. Definition of the standard basis of the polynomials of degree at most n

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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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