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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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526–550 of 715 nodes
  • Proof of several equivalences for nonsingular matrix#23644
  • QR decomposition#23630
  • Rank and mullity#23672
  • Rank and nullity#23882
  • Removing a linearly dependent vector from a set does not change the span of the set.#23982
  • R^n is a vector space.#24097
  • Row equivalence is an equivalence relation#23480
  • Row equivalent matrices have the same row space.#23881
  • Row equivalent matrices represent equivalent linear systems#23769
  • Row operations#23474
  • Row operations are given by multiplication by elementary matrices.#23559
  • Row operations do not necessarily preserve the column space.#23878
  • Scalar multiplication#23470
  • Similarity of matrices#23634
  • Similarity of matrices in an equivalence relation.#23636
  • Similar matrices have the same eigenvalues and the same characteristic polynomials.#23638
  • Spans#23804
  • Subspaces#23811
  • Subspaces associated to a linear transformation#24055
  • Subspaces associated to a matrix#23872
  • Switching two rows multiplies the determinant by -1.#23738
  • Symmetric matrices#23571
  • Symmetric matrices are square.#23573
  • Terminology#23754
  • The 0 scalar multiplied by any vector equals the 0 vector.#23960
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