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Elements

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Rank-nullity theorem — The rank-nullity theorem states that for any linear map where is finite-dimensional, the dimension of equals the sum of the map's rank and nullity.[1][2][3]

Observations

  • One
  • Two
  • Three

Every linear injection has a left-inverse.

Every linear surjection has a right-inverse.

Commentary

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There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.

Jean Dieudonné, Treatise on Analysis, Volume 1

We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.

Irving Kaplansky, in writing about Paul Halmos

Citations

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  1. ^ Axler (2015) p. 63, § 3.22
  2. ^ Katznelson & Katznelson (2008) p. 52, § 2.5.1
  3. ^ Valenza (1993) p. 71, § 4.3

Sources

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Textbooks

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  • Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.