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What is the time complexity of Gram Schmidt? It seems like this is at least O(k^2) or O(k^3) since you have to repeatedly subtract components of the first few vectors from all the remaining vectors, and each of these subtractions is a vector computation. Are there any more efficient algorithms for finding an orthonormal basis in a high dimensional space?


Is this algorithm still used in production anymore? It sounds like the other algorithm can be generalized to more cases.


What algorithms are used besides Gram-Schmidt?


What's an intuitive explanation for why <u2, e1> is the magnitude of the vector we should subtract from u2? I'm having trouble seeing why the dot product is being used here.


When talking about the disadvantage of the Gram-Schmidt algorithm when dealing with large number of vectors, you mentioned the QR decomposition method. How faster is the QR decomposition method comparing to the Gram-Schmidt method?


If the vectors are near parallel, what makes Graham Schmidt a poor choice? Would this be due to precision errors? Would an alternative approach involve finding a different basis, and then using Graham Schmidt on that?


I do not get what the relationship is between Gram-Schmidt and Orthonormal Basis. How do we choose between them?


This algorithm seems to be highly inefficient for a basis with a dimension of 3 or higher? Will we go over more efficient algorithms to find an orthonormal basis?