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  • Inner products in coordinate spaces

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  • Definition of inner/dot product on R^n
  • Definition of inner/dot product on C^n
  • The standard inner product on R^n is commutative.
  • The standard inner product on C^n is anticommutative.
  • The standard inner product on R^n commutes with (real) scalar multiplication.
  • The standard inner product on C^n commutes/anticommutes with scalar multiplication.
  • The standard inner product on R^n (or C^n) distributes over addition.
  • The standard inner product of a vector with itself is non-negative
  • The standard inner product of a vector with itself is 0 only for the 0 vector
  • The standard inner product on R^n can be written as the product of a vector and the transpose of a vector.
  • The standard inner product on C^n can be written as the product of a vector and the adjoint of a vector.
  • A matrix turns into its adjoint when moved to the other side of the standard inner product on C^n.
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  1. The standard inner product on R^n is commutative.

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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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