Elements
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
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
- ^ Axler (2015) p. 63, § 3.22
- ^ Katznelson & Katznelson (2008) p. 52, § 2.5.1
- ^ Valenza (1993) p. 71, § 4.3
Sources
Textbooks
- Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.
- Bogart, Kenneth P. (2000). Introductory Combinatorics. Harcourt Academic Press. ISBN 0-12-110830-9.
- Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7.
- Diestel, Reinhard (2017). Graph Theory (5 ed.). Springer. ISBN 978-3-662-53621-6.
- Gallian, Joseph A. (2012). Contemporary Abstract Algebra (8th ed.). Cengage. ISBN 978-1-133-59970-8.
- Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces (2nd ed.). Springer. ISBN 0-387-90093-4.
- Hefferon, Jim (2020). Linear Algebra (4th ed.). Orthogonal Publishing. ISBN 978-1-944325-11-4.
- Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9.
- Roman, Steven (2005). Advanced Linear Algebra (2nd ed.). Springer. ISBN 0-387-24766-1.
- Rotman, Joseph Jonah (1999). An Introduction to the Theory of Groups (4th ed.). Springer. ISBN 3-540-94285-8.
- Strang, Gilbert (2016). Introduction to Linear Algebra (5th ed.). Wellesley Cambridge Press. ISBN 978-0-9802327-7-6.
- Süli, Endre; Mayers, David (2011) [2003]. An Introduction to Numerical Analysis. Cambridge University Press. ISBN 978-0-521-00794-8.
- Tao, Terence (2017) [2014]. Analysis 1 (3rd ed.). Hindustan Book Agency. ISBN 978-93-80250-64-9.
- Tao, Terence (2017) [2014]. Analysis 2 (3rd ed.). Hindustan Book Agency. ISBN 978-93-80250-65-6.
- Trefethen, Lloyd Nicholas; Bau III, David (1997). Numerical Linear Algebra. SIAM. ISBN 978-0-898713-61-9.
- Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). Springer. ISBN 978-0-8218-4419-9.