Linear Algebra October 2017 comment.pdf - This comment is...

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This comment is about question 2 of the final exam. The statement of Gram-Schmidt theorem, as it was presented in lectures, is that for every set of vectors e_1,... e_n there exist orthogonal vectors f_1,...f_n such that the linear hull of f_1,...,f_k is the same as that of e_1,...,e_k for all k. In particular, it is true for k=n. The statement Gram-Schmidt theorem is not equivalent and does not reduce to the fact that one can choose an orthogonal base in a Euclidean vector space. Once again, it is essential that the flag of linear subspaces does not change when transiting from e_1,... e_k to f_1,...f_k for all k. One important consequence of Gram-Schmidt theorem is so called QR-decomposition of a matrix, where Q is
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