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  • Matrices as linear transformations

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  • Matrices act as a transformation by multiplying vectors
  • Matrix-vector multiplication is a linear transformation.
  • Definition of the standard matrix for a linear transformation
  • A linear transformation is given by a matrix whose columns are the images of the standard basis vectors
  • A linear transformation is given by a matrix with respect to a given basis.
  • Definition of nilpotent linear transformation
  • Definition of matrix representation of a linear transformation
  • Matrix representation of a composition of linear transformations is given by a matrix product
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  1. Definition of nilpotent linear transformation

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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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