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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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  • Geometric properties of R^n (or C^n)#23795
  • Hermitian matrices#23596
  • Hermitian matrices have real eigenvalues.#23711
  • Homogeneous linear systems are consistent.#23434
  • If A and B are n-by-n matrices#23729
  • If A is a matrix#23679
  • If a matrix has both a left and a right inverse#23519
  • If a set of vectors contains the 0 vector#23825
  • If a set of vectors in R^n (or C^n) contains more than n elements#23827
  • If a space is the direct sum of invariant subspaces#24053
  • If a square matrix has a one-sided inverse#23520
  • If a vector space has dimension n#23838
  • If B is a basis containing b and the b coordinate of c is nonzero#23990
  • If the product of a vector and a scalar is 0#23962
  • If two finite dimensional subspaces have the same dimension and one is contained in the other#24011
  • If two matrices have equal products with all vectors#23484
  • Inner products#24087
  • Inner products in coordinate spaces#23894
  • Inverse#23514
  • Isomorphic vector spaces have the same dimension.#24004
  • Isomorphism#24000
  • Least squares#23945
  • Linear algebra#23410
  • Linear combinations#23799
  • Linear (in)dependence#23815
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  • Aspen Summer 2016 - Entanglement Matters#1213
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