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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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  • Abstract vector spaces#23953
  • A consistent system with more variables than equations has infinitely many solutions.#23440
  • Adding a multiple of one row to another row does not change the determinant.#23737
  • Addition#23461
  • A determinant is a multilinear function#23736
  • A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns.#23615
  • Adjoining an element not in the span of a linearly independent set gives another linearly independent set.#23983
  • A homogeneous system has a nontrivial solution if and only if it has a free variable.#23437
  • A homogeneous system with more variables than equations has infinitely many solutions.#23441
  • Algebraic properties of R^n (or C^n)#23777
  • Algorithm for computing an LU decomposition#23626
  • A linear system is equivalent to a matrix equation.#23753
  • A linear system is equivalent to a vector equation.#23752
  • A linear transformation has a representation as an upper triangular matrix.#24076
  • A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.#24075
  • A linear transformation is determined by its action on a basis.#23866
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.#24054
  • A linear transformation is given by a matrix whose columns are the images of the standard basis vectors#23858
  • A linear transformation is given by a matrix with respect to a given basis.#23859
  • A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces#24069
  • A linear transformation is injective on its generalized range space.#24040
  • A linear transformation is invertible if and only if it is injective and surjective#24067
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.#24066
  • A linear transformation is surjective if and only if the columns of its matrix span the codomain.#23885
  • A linear transformation is surjective if and only if the image of a basis is a spanning set#24038
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