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"$$\n",
"\\newcommand{\\vb}{\\mathbf{b}}\n",
"\\newcommand{\\vc}{\\mathbf{c}}\n",
"\\newcommand{\\vx}{\\mathbf{x}}\n",
"\\newcommand{\\mA}{\\mathbf{A}}\n",
"\\newcommand{\\mL}{\\mathbf{L}}\n",
"\\newcommand{\\mU}{\\mathbf{U}}\n",
"\\newcommand{\\mP}{\\mathbf{P}}\n",
"\\newcommand{\\mI}{\\mathbf{I}}\n",
"$$\n",
"\n",
"1. Given least squares problem $\\mA\\vx\\approx\\vb$, suppose that the right hand side $\\vb$ matches one of the columns of $\\mA$. Does this problem have a unique solution? Can the problem be solved with residual $\\mA=0$? Do these answers change when $\\vb\\notin\\mathrm{span}\\mA$?
\n",
"\n",
"2. Suppose that we just performed a LU factorization of a matrix $\\mA\\in\\mathbb{R}^{n\\times n}$ with partial pivoting, resulting in the factorization $\\mA=\\mP\\mL\\mU$. Is it possible to efficiently compute the determinant of $\\mA$ given this information? (You may need to review rules for computing the determinant of products and orthogonal and triangular matrices). Approximately how many FLOPs will this computation consume?\n",
"\n",
" Assuming we find that $\\det{\\mA}=\\varepsilon$ (where $\\varepsilon$ is a tiny number, e.g. $10^{-300}$), should this make us nervous about potential numerical difficulties when solving linear systems involving $\\mA$?
"
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