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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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326–350 of 715 nodes
  • Definition of weights in a linear combination of vectors#23802
  • Description of a basis for the null space of a matrix from the reduced row-echelon form.#23870
  • Description of a spanning set for the null space of a matrix from the reduced row-echelon form.#23869
  • Description of the Gram-Schmidt process#23935
  • Determinants#23714
  • Determinants and operations on matrices#23723
  • Determinants axiomatically#23732
  • Determine if a particular set of vectors in R^3 in linearly independent#23818
  • Determine if a particular set of vectors spans R^3#23807
  • Determine if a particular vector is in the span of a set of vectors#23808
  • Determine if a particular vector is in the span of a set of vectors in R^2#23809
  • Determine if a particular vector is in the span of a set of vectors in R^3#23810
  • Dimension#23836
  • Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.#23712
  • Each vector can be written uniquely as a linear combination of vectors from a given basis.#23987
  • Echelon matrices#23536
  • Eigenspaces#23683
  • Eigenvalues and eigenvectors#23680
  • Eigenvalues and operations on matrices#23688
  • Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal.#23574
  • Eigenvectors with distinct eigenvalues are linearly independent.#23694
  • Elementary matrices#23556
  • Elementary matrices are invertible/nonsingular.#23560
  • Equation operations on a linear system give an equivalent system.#23425
  • Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.#24086
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