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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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  • Definition of matrix-scalar multiplication#23471
  • Definition of matrix-vector product#23483
  • Definition of m by n matrix#23452
  • Definition of minimal polynomial of a linear transformation#24045
  • Definition of minimal polynomial of a matrix#23703
  • Definition of multilinear function#23733
  • Definition of nilpotent linear transformation#23860
  • Definition of nilpotent matrix#23581
  • Definition of nonsingular matrix: matrix is invertible#23641
  • Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solution#23642
  • Definition of nontrivial solution to a homogeneous linear system of equations#23435
  • Definition of normal matrix#23600
  • Definition of norm/length of a vector#23908
  • Definition of (not necessarily orthogonal) projection onto a component of a direct sum#23940
  • Definition of nullity of a linear transformation#23884
  • Definition of nullity of a matrix#23675
  • Definition of one-to-one/injective linear transformation#23845
  • Definition of onto/surjective linear transformation#23844
  • Definition of orthogonal basis of a (sub)space#23928
  • Definition of orthogonal complement of a subspace#23920
  • Definition of orthogonally diagonalizable matrix#23577
  • Definition of orthogonal matrix#23586
  • Definition of (orthogonal) projection of one vector onto another vector#23938
  • Definition of (orthogonal) projection onto a subspace#23941
  • Definition of orthogonal set of vectors#23925
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