Advanced Linear Algebra - Advanced Linear Algebra, Lecture 1.5: Dual vector spaces
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.
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Advanced Linear Algebra, Lecture 1.5: Dual vector spaces
The dual of a vector space X over K is the space X' of all linear scalar functions from X to K, which are also called co-vectors or dual vectors. When dim(X)=n is finite, then X and X' are isomorphic. We can think about vectors as length-n column vectors, and dual vectors as length-n row vectors. The function l(x) is simply the scalar product of these, so we can denote it as (l,x)=l(x). We conclude with an example of an infinite-dimensional vector space that has a dual vector that is not of this form.
Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html
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