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  • Linear systems and matrices
  • Linear transformations

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  • Eigenvalues and eigenvectors
  • A linear system is equivalent to a vector equation.
  • A linear system is equivalent to a matrix equation.
  • Terminology
  • Using matrices to solve linear systems
  • Matrix equations
  • Row equivalent matrices represent equivalent linear systems
  • Linear systems and echelon matrices
  • 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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  • Definition of augmented matrix (of a linear system)
  • Definition of coefficient matrix of a linear system
  • Definition of constant vector of a linear system
  • Definition of solution vector of a linear system
  • Definition of matrix representation of a linear system
  • Definition of domain of a linear transformation
  • Definition of codomain of a linear transformation
  • Definition of image (of a point) under a linear transformation
  • Definition of pre-image (of a point) under a linear transformation
  • Definition of onto/surjective linear transformation
  • Definition of one-to-one/injective linear transformation
  • Definition of range of linear transformation
  • Definition of kernel of linear transformation
  • Definition of invertible linear transformation
  • Definition of inverse of a linear transformation
  • Non-example of a linear transformation
  • Definition of linear transformation/homomorphism
  • Definition of identity linear transformation
  • Definition of sum of linear transformations
  • The sum of linear transformations is a linear transformation
  • Definition of scalar multiple of a linear transformation
  • A scalar multiple of a linear transformation is a linear transformation
  • Definition of pre-image of linear transformation
  • Definition of range of a linear transformation
  • Definition of invertible/nonsingular linear transformation
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  • Definition of orthogonal subspaces#24094
  • Definition of orthogonal vectors#23916
  • Definition of orthonormal basis of a (sub)space#23929
  • Definition of orthonormal set of vectors#23926
  • Definition of parallel vectors#23917
  • Definition of permutation matrix#23555
  • Definition of pivot#23476
  • Definition of pivot column#23477
  • Definition of pivot position#23478
  • Definition of positive-definite matrix#23713
  • Definition of pre-image (of a point) under a linear transformation#23843
  • Definition of pre-image of linear transformation#24019
  • Definition of QR decomposition#23631
  • Definition of quadratic form#24122
  • Definition of range of a linear transformation#24020
  • Definition of range of linear transformation#23846
  • Definition of rank factorization of a matrix#23628
  • Definition of rank of a linear transformation#23883
  • Definition of rank of a matrix#23674
  • Definition of rational form#23621
  • Definition of reduced LU decomposition#23627
  • Definition of reduced row echelon form of a matrix#23544
  • Definition of right inverse of a matrix#23516
  • Definition of R^n (or C^n)#23781
  • Definition of (row) echelon form of a matrix#23542
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