Advanced Linear Algebra - Advanced Linear Algebra, Lecture 1.6: Annihilators
Unlock the Power of Vector Spaces: Master Advanced Linear Algebra with Professor Macauley. Dive deep into theory and applications, from eigenvectors to spectral theorems. Enhance your mathematical prowess and transform complex problems into elegant solutions.
5.0(4)
43 learners
What you'll learn
Understand key concepts of vector spaces, including spanning, independence, and bases.
Analyze the role of eigenvalues, eigenvectors, and the spectral theorem in linear mappings.
Apply the Gram-Schmidt process and orthogonal projection in various contexts.
Evaluate the properties and applications of quadratic forms and spectral resolutions.
Advanced Linear Algebra, Lecture 1.6: Annihilators
The annihilator of a subspace Y of X is the collection of all linear scalar functions that vanish on Y. Its dimension, as a subspace of the dual X', is called the codimension of Y, denoted codim(Y). We will show that dim(Y)+codim(Y)=dim(X), and then why the annihilator of the annihilator of Y is simply the subspace Y itself, under the canonical identification of X with its double dual, X''. Finally, we define the annihilator of an arbitrary subset S, and show how that is just the annihilator of its span.
Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html
Continue this lesson in the app
Install CourseHive on Android or iOS to keep learning while you move.
