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homework3.ipynb

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+    "### Neşat Dereli\n",
+    "\n",
+    "## 3.6.a\n",
+    "\n",
+    "(1) $\\|x\\|' = 0$ if and only if x is zero vector\n",
+    "\n",
+    "\\begin{eqnarray}\n",
+    "&\\|x\\|' &= &sup_{\\|y\\|=1} |y^*x| \\geq 0 \\qquad \\text{ since absolute value}\\\\\n",
+    "&\\|x\\| &= &0 \\Leftrightarrow \\text{x is zero vector}\\\\\n",
+    "&\\|x\\|' &= &0 \\Leftrightarrow \\|x\\| = 0\\\\\n",
+    "&\\|x\\|' &= &0 \\Leftrightarrow \\text{x is zero vector}\\\\\n",
+    "\\end{eqnarray}\n",
+    "\n",
+    "(2) $\\|x + z\\|' \\leq \\|x\\|' + \\|z\\|'$\n",
+    "\n",
+    "\\begin{eqnarray}\n",
+    "&\\|x + z\\|' &= &sup_{\\|y\\|=1} |y^*(x+z)|\\\\\n",
+    "& &= &sup_{\\|y\\|=1} |y^*x + y^*z)|\\\\\n",
+    "& &\\leq &sup_{\\|y\\|=1} | y^*x| + sup_{\\|y\\|=1} |y^*z |\\\\\n",
+    "& &\\leq &\\|x\\|' + \\|z\\|'\\\\\n",
+    "\\end{eqnarray}\n",
+    "\n",
+    "(3) $\\|ax\\|' = |a| \\|x\\|'$\n",
+    "\n",
+    "\\begin{eqnarray}\n",
+    "&\\|ax\\|' &= &sup_{\\|y\\|=1} |y^*ax|\\\\\n",
+    "& &= &|a|sup_{\\|y\\|=1} |y^*x|\\\\\n",
+    "& &= &sup_{\\|y\\|=1} |a||y^*x|\\\\\n",
+    "& &= &|a| \\|x\\|'\\\\\n",
+    "\\end{eqnarray}\n",
+    "\n",
+    "\n",
+    "\n",
+    "## 4.5\n",
+    "\n",
+    "\\begin{eqnarray}\n",
+    "A \\in \\mathbb{R}^{mxn} \\Rightarrow A^*A = A^TA \\in \\mathbb{R}^{nxn}\\\\\n",
+    "A^TA = V\\Sigma^2V^T\\\\\n",
+    "A^*A = V\\Sigma^2V^*\\\\\n",
+    "\\Sigma^2 \\in \\mathbb{R}^{nxn}\\\\\n",
+    "A^TA = A^*A \\Rightarrow V\\Sigma^2V^T = V\\Sigma^2V^*\\\\\n",
+    "V^T = V^* \\Rightarrow V \\in \\mathbb{R}^{nxn}\n",
+    "\\text{ }\\\\\n",
+    "AA^T = U^T\\Sigma^2U\\\\\n",
+    "AA^* = U^*\\Sigma^2U\\\\\n",
+    "\\Sigma^2 \\in \\mathbb{R}^{mxm}\\\\\n",
+    "AA^T = AA^* \\Rightarrow U^T\\Sigma^2U = U^*\\Sigma^2U\\\\\n",
+    "U^T = U^* \\Rightarrow U \\in \\mathbb{R}^{mxm}\n",
+    "\\end{eqnarray}\n",
+    "\n",
+    "\n",
+    "## 5.3\n"
+   ]
+  }
+ ],
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+     "url": "https://github.com/Calysto/octave_kernel"
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+     "text": "MetaKernel Magics",
+     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
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