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Algorithm for solving linear equations if interested only in the first component

Follow the Gaussian elimination and manipulate the matrix $A$ such that in one row only the coefficient of $x_1$ survives!
codeR's user avatar
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Matrix Multiplication using LU decomposition

Nevermind, I didn't think about standard matrix multiplication with a vector which is the same https://stackoverflow.com/questions/35182616/the-runtime-complexity-of-the-product-nxn-matrix-time-n-...
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