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  • Subspaces associated to a linear transformation

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  • The range/image of a linear transformation is a subspace.
  • The inverse image of a subspace under a linear transformation is a subspace.
  • Definition of kernel/null space of linear transformation
  • The kernel/null space of a linear transformation is a subspace
  • Definition of generalized kernel/null space of linear transformation
  • Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.
  • The kernels of powers of a linear transformation form an ascending chain
  • The range spaces of powers of a linear transformation form a descending chain
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  1. Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.

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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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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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