Advanced Linear Algebra - Advanced Linear Algebra, Lecture 2.5: The transpose of a linear map

Advanced Linear Algebra, Lecture 2.5: The transpose of a linear map If T is a linear map from X to U, then this induces a linear map T' from U' to X', that "precomposes" linear scalar functions with T. That is, it maps ℓ to ℓT. In our scalar product notation, this means that (T'ℓ,x)=(ℓ,Tx). We unpack this definition in the context of systems of equation, and how it relations to column and row vectors. We give a very straightforward proof that the annihilator of the range of T is the nullspace of its transpose. This is the abstract version of the fact that the column space is orthogonal to the left nullspace, and that the row space is orthogonal to the null space. An easy corollary is that the range T and its transpose T' have the same dimension, which is the abstract version of the fact that the column space and row space have the same dimension, the rank of T. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html