wip midterm

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}