Advanced Linear Algebra - Advanced Linear Algebra, Lecture 2.1: Rank and nullity
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.
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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 2.1: Rank and nullity
If T is a linear map from X to U, then the rank is the dimension of the image (a subspace of U), and the nullity is the dimension of the nullspace (a subspace of X). A fundamental result of finite-dimensional vector spaces is that these numbers add up to dim X. After proving this, we look at several easy special cases, and specialize them to systems of equation. For example, if we have n equations on n variables, and the null space of T is trivial, then the inhomogeneous system Tx=u always has a unique solution.
Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html
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