Advanced Linear Algebra - Advanced Linear Algebra, Lecture 5.3: Gram-Schmidt and orthogonal projection

Advanced Linear Algebra, Lecture 5.3: Gram-Schmidt and orthogonal projection We begin this lecture with the Gram-Schmidt process, which takes an arbitrary basis and creates an orthonormal basis. The matrix form of this construction is called the QR factorization: if we put the columns of this old basis into a matrix A, then we can write A=QR, where Q is the matrix whose columns is the orthonormal basis, and the entries in (the upper-triangular matrix) R are the projections of each old basis vector into the new coordinate system. After that, we canonically identify a space X with its dual X', generalizing our earlier idea of thinking of dual vectors as row vectors, and ordinary vectors as column vectors. Finally, we define the orthogonal complement of a subspace Y, and the orthogonal projection onto that subspace. We prove that in any inner product space, this sends x to the vector y in Y that minimizes ||x-y||. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html