Advanced Linear Algebra - Advanced Linear Algebra, Lecture 5.4: Adjoints

Advanced Linear Algebra, Lecture 5.4: Adjoints An inner product space X can be naturally identified with its dual via a map sending y to (-,y). Under this identification, the transpose of linear map from X to U is simply a map A' between the dual spaces U' and X', that sends ℓ=(-,u) to m=(-,y). This naturally defines a map called the adjoint, denoted A*, from U to X, sending u to y. Unlike the transpose, the adjoint depends on the inner product structure. However, it has many similar properties, and is characterized by the quality (x,A*u)=(Ax,u). We will conclude by showing how two basic properties of matrices: "the orthogonal complement of the row space is the nullspace", and "the orthogonal complement of the column space is the left nullspace", are characterized by adjoints. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html