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xg60g!zJ~Qd;})bT$wTuRPKyRFey%kx{`c7U#MtmtAPfd)kGBUQkwkwl&_7lkp!@&; diff --git a/midterm/midterm.tex b/midterm/midterm.tex index 92ac1fe..415e779 100644 --- a/midterm/midterm.tex +++ b/midterm/midterm.tex @@ -37,7 +37,8 @@ $$1|101010101010|010_F = -(1+2^{-1} +2^{-3} +2^{-5} +2^{-7} +2^{-9} + 2^{-11}) * $$1|0000 0000 0000|001_F = 2^{-3}_{10} = 0.125_{10}$$ \subsection*{Point d)} -$$1|1111 1111 1111|111_F = (1 + 2^{-1}+ 2^{-2} + 2^{-3} +2^{-4} + 2^{-5} +2^{-6} + 2^{-7} + 2^{-8} + 2^{-9}+ 2^{-10}+ 2^{-11}+ 2^{-12}) * {2^3}_{10} = $$ +$$1|1111 1111 1111|111_F = $$ +$$=(1 + 2^{-1}+ 2^{-2} + 2^{-3} +2^{-4} + 2^{-5} +2^{-6} + 2^{-7} + 2^{-8} + 2^{-9}+ 2^{-10}+ 2^{-11}+ 2^{-12}) * {2^3}_{10} = $$ $$ = 15.998046875_{10}$$ \subsection*{Point e)} @@ -169,7 +170,7 @@ u_4 = \end{bmatrix} \] \[ -\l_5 = +l_5 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ \end{bmatrix}, @@ -196,7 +197,7 @@ u_5 = \] \subsection*{Point b)} \[ -\l_1 = +l_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ \end{bmatrix}, @@ -214,7 +215,7 @@ e^{T}_1 = \end{bmatrix} \] \[ -\l_2 = +l_2 = \begin{bmatrix} 0 \\ 1 \\ 2 \\ 3 \\ 4 \\ \end{bmatrix}, @@ -232,7 +233,7 @@ e^{T}_2 = \end{bmatrix} \] \[ -\l_3 = +l_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 2 \\ 3 \\ \end{bmatrix}, @@ -250,7 +251,7 @@ e^{T}_3 = \end{bmatrix} \] \[ -\l_4 = +l_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \\ 2 \\ \end{bmatrix}, @@ -267,5 +268,72 @@ e^{T}_4 = 0 & 0 & 0 & -2 & 1 \\ \end{bmatrix} \] -\subsection{Point c)} - \end{document} +\subsection*{Point c)} +$$Ly = b$$ +\[ + L = + \begin{bmatrix} + 1 & 0 & 0 & 0 & 0 \\ + 2 & 1 & 0 & 0 & 0 \\ + 3 & 2 & 1 & 0 & 0 \\ + 4 & 3 & 2 & 1 & 0 \\ + 5 & 4 & 3 & 2 & 1 \\ + \end{bmatrix}, +b = + \begin{bmatrix} + 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ + \end{bmatrix} + \] + \[ + L = + \begin{bmatrix} + 1 & 0 & 0 & 0 & 0 \\ + 2 & 1 & 0 & 0 & 0 \\ + 3 & 2 & 1 & 0 & 0 \\ + 4 & 3 & 2 & 1 & 0 \\ + 5 & 4 & 3 & 2 & 1 \\ + \end{bmatrix}, +b = + \begin{bmatrix} + 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ + \end{bmatrix} + \] + $$y_1 = \frac{1}{1} = 1$$ + $$y_2 = \frac{2 - 2 \cdot 1}{1} = 0$$ + $$y_3 = \frac{3 - 3 \cdot 1 - 2 \cdot 0}{1} = 0$$ + $$y_4 = \frac{4 - 4 \cdot 1 - 3 \cdot 0 - 2 \cdot 0}{1} = 0$$ + $$y_5 = \frac{5 - 5 \cdot 1 - 4 \cdot 0 - 3 \cdot 0 - 2 \cdot 0}{1} = 0$$ + \[ +y = + \begin{bmatrix} + 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{bmatrix} + \] + \subsection*{Point d)} +$$Ux = y$$ +\[ + U = + \begin{bmatrix} + 1 & 1 & 1 & 1 & 1 \\ + 0 & 2 & 2 & 2 & 2 \\ + 0 & 0 & 3 & 3 & 3 \\ + 0 & 0 & 0 & 4 & 4 \\ + 0 & 0 & 0 & 0 & 5 \\ + \end{bmatrix} +y = + \begin{bmatrix} + 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{bmatrix} + \] + $$x_5 = \frac{0}{5} = 0$$ + $$x_4 = \frac{0 - 4 \cdot 0}{4} = 0$$ + $$x_3 = \frac{0 - 3 \cdot 0 - 3 \cdot 0}{3} = 0$$ + $$x_2 = \frac{0 - 2 \cdot 0 - 2 \cdot 0 - 2 \cdot 0}{2} = 0$$ + $$x_1 = \frac{0 - 1 \cdot 0 - 1 \cdot 0 - 1 \cdot 0 - 1 \cdot }{1} = 0$$ + \[ +x = + \begin{bmatrix} + 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{bmatrix} + \] +\end{document} \ No newline at end of file