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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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  • Triangular matrices#23561
  • Two matrices of the same size are equivalent if and only if they have the same rank.#23607
  • Two vectors are orthogonal if and only if the Pythagorean Theorem holds.#23918
  • Unitary matrices#23587
  • Unitary matrices are invertible.#23589
  • Unitary matrices have orthogonal (orthonormal) rows/columns.#23590
  • Unitary matrices preserve inner products.#23591
  • Unitary matrices preserve orthogonal (orthonormal) bases.#23592
  • Using matrices to solve linear systems#23760
  • Vector space isomorphism is an equivalence relation.#24003
  • Vector spaces#23775
  • Vector spaces with the same dimension are isomprphic.#24005
  • Vector sum/addition interpreted geometrically in R^n (or C^n)#23797
  • Vector sum/addition is commutative and associative#23787
  • Visualise a linear transformation on R^2 by looking at the image of a region#23852
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