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  • Linear transformations

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  • Eigenvalues and eigenvectors
  • The matrix representation of a sum of linear transformations is the sum of the matrices.
  • A linear transformation is invertible if and only if it is injective and surjective
  • Subspaces associated to a linear transformation
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
  • The preimage of a vector is a translation of the kernel of the linear transformation
  • The matrix representation of a composition of linear transformations is the product of the 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
  • Terminology
  • Composition
  • 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 image of a spanning set spans the range
  • 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
  • The rank plus the nullity of a linear transformation equals the dimension of the domain.
  • Linear maps preserve dependence of lists, with image-set caveats
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
  • 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 of an inverse is the inverse of the matrix representation
  • Eigenvalues agree and eigenvectors correspond through coordinates
  • Triangularization requires the characteristic polynomial to split
  • 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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  • Example of vector-scalar multiplication in R^2#23794
  • Example of writing a given vector in R^3 as a linear combination of given vectors#23803
  • Examples#23886
  • Examples of vector spaces#24095
  • Factorization of matrices#23622
  • Finite-dimensional Fitting decomposition into generalized kernel and range#24061
  • Finite-dimensional vector spaces of equal dimension over the same field are isomorphic#24005
  • F^n is a vector space.#24099
  • Formula for computing the least squares solution to a linear system#23949
  • Formula for computing the least squares solution to a linear system.#23948
  • Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.#23933
  • Formula for the coordinates of the projection of a vector onto a subspace#23944
  • Formula for the determinant of a 2-by-2 matrix.#23742
  • Formula for the determinant of a 3-by-3 matrix.#23743
  • Formula for the inverse of a 2-by-2 matrix.#23521
  • Formula for the least-squares linear fit to 2-dimensional data#23952
  • Formula for the (orthogonal) projection of one vector onto another vector#23939
  • Formula for the spectral decomposition for a symmetric matrix#23576
  • For n-by-n invertible matrices A and B#23523
  • Gaussian elimination as a method to solve a linear system#23421
  • Gauss-Jordan procedure to put a matrix into reduced row echelon form#23540
  • Geometric description of span of a set of vectors in R^n (or C^n)#23806
  • Geometric picture of a 2-by-2 linear system#23428
  • Geometric picture of a 3-by-3 linear system#23430
  • Geometric picture of the solution set of a linear equation in 3 unknowns#23429
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Container for Linear Algebra#24123
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