Linear Algebra and Matrix Theory
Corrections and Additions
Fall 2026

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Corrections

The corrections are for Sheldon Axler, Linear Algebra Done Right, 4th edition.

Chapter 3. Most of this is valid over any field.
Page 52. Many, possibly most, mathematicians call a linear map "a linear transformation".
Definition 3.11. Most mathematicians call his "null space" the "kernel" of the linear transformation, written ker T.
"Range" and "image" are synonyms, both widely used. One often sees im T for the range/image.
Definition 3.29. His matrix notation is bad. Standard notation for an element of matrix A is a_i,j: lower case elements for upper case matrices. The standard meaning of A_i,j is the cofactor of a_i,j, which is entirely different.
Definition 3.31. Axler is right about M(T). It's important to remember a linear map does not have a matrix, unless ordered bases are specified.
That even applies to maps F^m → Fⁿ, where the vector spaces have standard bases; there is no rule that you have to use the standard bases.
Notations 3.44 are highly nonstandard. (Just a warning.)