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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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  • The 0 vector is unique.#23958
  • The 0 vector multiplied by any scalar equals the 0 vector.#23961
  • The additive inverse of a vector equals the vector multiplied by -1.#23963
  • The additive inverse of a vector is called the negative of the vector.#23956
  • The additive inverse of a vector is unique.#23959
  • The adjoint of a matrix-scalar product is the product of the adjoint and the conjugate.#23508
  • The adjoint of a product of matrices is the product of the adjoints in reverse order.#23513
  • The adjoint of a sum is the sum of the adjoints.#23507
  • The Cauchy-Schwartz inequality#23913
  • The Cauchy-Schwarz inequality#24092
  • The Cayley-Hamilton theorem for a linear transformation#24046
  • The Cayley-Hamilton theorem for a matrix.#23707
  • The change of coordinates matrix between two bases exists and is unique#23996
  • The characteristic polynomial applied to the matrix gives the 0 matrix.#23706
  • The column space of a matrix is a vector space#24109
  • The column space of an m-by-n matrix is a subspace of R^m (or C^m)#23876
  • The composition of injective linear transformations is injective#24025
  • The composition of invertible linear transformations is invertible#24027
  • The composition of linear transformations is a linear transformation#24024
  • The composition of surjective linear transformations is surjective#24026
  • The condition number of matrix measures how close it is to being singular#23750
  • The conjugate of a matrix-scalar product is the product of the conjugates.#23468
  • The conjugate of a product of matrices is the product of the conjugates.#23469
  • The conjugate of a sum of vectors in C^n is the sum of the conjugates#23791
  • The conjugate of the sum of matrices is the sum of the conjugates.#23467
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