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Elements

Rank-nullity theorem — The rank-nullity theorem states that for any linear map $\Phi :{\mathcal {V}}\to {\mathcal {W}}$ where ${\mathcal {V}}$ is finite-dimensional, the dimension of ${\mathcal {V}}$ equals the sum of the map's rank and nullity.

$$\dim {\mathcal {V}}=\dim \operatorname {img} \Phi +\dim \ker \Phi$$

$$\dim {\mathcal {V}}=\operatorname {rank} \Phi +\operatorname {nullity} \Phi$$

Observations

• One
• Two
• Three

Every linear injection has a left-inverse.

$$\Phi _{\mathrm {L} }^{-1}=(\Phi ^{\intercal }\Phi )^{-1}\Phi ^{\intercal }$$

Every linear surjection has a right-inverse.

$$\Phi _{\mathrm {R} }^{-1}=\Phi ^{\intercal }(\Phi \Phi ^{\intercal })^{-1}$$

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

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.