Advanced Linear Algebra - Advanced Linear Algebra, Lecture 1.4: Quotient 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.
Advanced Linear Algebra, Lecture 1.4: Quotient spaces
If two vectors x and z differ by an element y in a subspace Y, then we say that x≡z (mod Y). This defines an equivalence relation, and the equivalence classes form a vector space called the quotient space of X modulo Y, and denoted X/Y. We define addition and scalar multiplication in this space and show that it is well-defined, as well as discuss what that means. We give some examples of quotient spaces, and prove several basic theorems, such as dim(Y)+dim(X/Y)=dim(Z), and dim(U+V)=dim(U)+dim(V)-dim(U⋂V).
Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html
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