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  • Linear transformations

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  • Eigenvalues and eigenvectors
  • The matrix representation of a sum of linear transformations is the sum of the matrices.
  • A linear transformation is invertible if and only if it is injective and surjective
  • Subspaces associated to a linear transformation
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
  • The preimage of a vector is a translation of the kernel of the linear transformation
  • The matrix representation of a composition of linear transformations is the product of the matrices.
  • Geometric properties of linear transformations
  • Matrices as linear transformations
  • Basic properties of linear transformations
  • Description of a spanning set for the null space of a matrix from the reduced row-echelon form.
  • Description of a basis for the null space of a matrix from the reduced row-echelon form.
  • The nonzero rows of an echelon form of a matrix are linearly independent.
  • Subspaces associated to a matrix
  • Rank and nullity
  • Examples
  • Terminology
  • Composition
  • The image of a linearly independent set under an injective linear transformation is linearly independent.
  • The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.
  • The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.
  • A linear transformation is surjective if and only if the rank equals the dimension of the codomain.
  • The range of a linear transformation is a subspace
  • The image of a spanning set spans the range
  • A linear transformation is surjective if and only if the image of a basis is a spanning set
  • Definition of generalized range space of a linear transformation
  • A linear transformation is injective on its generalized range space.
  • Definition of diagonalizable linear transformation
  • The rank plus the nullity of a linear transformation equals the dimension of the domain.
  • Linear maps preserve dependence of lists, with image-set caveats
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
  • Definition of matrix representation of a linear transformation with respect to bases of the spaces
  • A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces
  • Definition of matrix representation of a linear transformation from a vector space to itself
  • The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.
  • The matrix of an inverse is the inverse of the matrix representation
  • Eigenvalues agree and eigenvectors correspond through coordinates
  • Triangularization requires the characteristic polynomial to split
  • Equivalence theorems for injective transformations

Children8
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  • Definition of composition of linear transformations
  • The composition of linear transformations is a linear transformation
  • The composition of injective linear transformations is injective
  • The composition of surjective linear transformations is surjective
  • The composition of invertible linear transformations is invertible
  • The inverse of a linear transformation is a linear transformation
  • The inverse of the inverse of a linear transformation is the original linear transformation
  • The inverse of a composition reverses the order
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  • Axioms of a vector space#23798
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