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  • Linear transformations

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  • Eigenvalues and eigenvectors
  • Terminology
  • 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
  • Composition
  • The preimage of a vector is a translation of the kernel of the linear transformation
  • 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 the image of a spanning set is a spanning set for the range space
  • 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
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
  • Subspaces associated to a linear transformation
  • The rank plus the nullity of a linear transformation equals the dimension of the domain.
  • The image of a linearly dependent set under a linear transformation is linearly dependent.
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
  • A linear transformation is invertible if and only if it is injective and surjective
  • 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 representation of a sum of linear transformations is the sum of the matrices.
  • The matrix representation of a composition of linear transformations is the product of the matrices.
  • The matrix representation of the inverse of linear transformations is the inverse of the matricix.
  • A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.
  • A linear transformation has a representation as an upper triangular matrix.
  • Equivalence theorems for injective transformations
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  1. A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.

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501–525 of 715 nodes
  • Matrix transpose commutes with matrix inverse.#23526
  • Matrix transpose is an involution.#23504
  • Matrix-vector multiplication is a linear transformation.#23856
  • Matrix-vector product is associative#23486
  • Matrix-vector products#23482
  • Multiplication#23487
  • Multiplication by a change of coordinates matrix converts representations for different bases.#23997
  • Multiplication by a Hermitian matrix commutes with the standard inner product on C^n.#23598
  • Multiplication of block/partitioned matrices#23569
  • Multiplicity#23695
  • Multiplying a row by a scalar multiplies the determinant by that scalar.#23739
  • Nilpotent matrices#23580
  • Non-example of a linear transformation#23850
  • Nonsingular matrices and equivalences#23640
  • Normal matrices#23599
  • Norm and length#23907
  • Notation for entry of matrix#23450
  • Notation for the set of m by n matrices#23453
  • Operations on matrices#23460
  • Orthogonality#23914
  • Orthogonality and projection#23893
  • Parametric form of the solution set of a system of linear equations#23419
  • Parametric vector form of the solution set of a system of linear equations#23420
  • Particular types of matrices#23535
  • Projection#23937
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