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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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  • An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).#23616
  • An n-by-n matrix nas n (complex) eigenvalues#23698
  • An n-by-n matrix with n distinct eigenvalues is diagonalizable.#23617
  • A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations#23965
  • A nonsingular matrix can be written as a product of elementary matrices.#23558
  • An orthogonal set of nonzero vectors is linearly independent.#23927
  • Any linearly independent set can be expanded to a basis for the (sub)space#23984
  • Any matrix times the 0 matrix equals the 0 matrix.#23491
  • Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.#24061
  • Application Leontief input-output analysis#24118
  • Applications#24113
  • Applications of band matrices#24116
  • Applications to cubic spline#24120
  • Applications to differential equations#24114
  • Applications to error-correcting code#24117
  • Applications to Markov chains#24115
  • Applications to voting and social choice#24119
  • A scalar multiple of a linear transformation is a linear transformation#24018
  • A set is a basis if each vector can be written uniquely as a linear combination.#23988
  • A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.#23981
  • A set of nonzero vectors contains (as a subset) a basis for its span.#23985
  • A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.#23826
  • A set of vectors containing fewer elements than the dimension of the space cannot span#23979
  • A set of vectors containing more elements than the dimension of the space must be linearly dependent#24012
  • A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.#23823
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