8-dot-product-and-norm.ipynb

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    "# Dot product and norm\n",
    "\n",
    "Recall that the dot product of two vectors $x,y \\in \\mathbb{R}^{n}$ is\n",
    "\n",
    "$x \\cdot y = x_{1}y_{1} + x_{2}y_{2} + \\dots + x_{n}y_{n} = x^{T}y$,\n",
    "\n",
    "and the norm of a vector $x$ is \n",
    "\n",
    "$\\| x \\| = \\sqrt{x \\cdot x}.$\n",
    "\n",
    "Actually `numpy` has both functions built in (try googling by yourself ;D ), but we shall write our own functions just to learn more."
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   "source": [
    "import numpy as np"
   ]
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   "source": [
    "def dot(x,y):\n",
    "    d = x.T@y\n",
    "    return d\n",
    "\n",
    "def norm(x):\n",
    "    n = np.sqrt(dot(x,x))\n",
    "    return n"
   ]
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   "source": [
    "x = np.array([[3, -2, 0, 3]]).T\n",
    "y = np.array([[2, 0, 1, -2]]).T"
   ]
  },
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     "text": [
      "x.y = [[0]]\n"
     ]
    }
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   "source": [
    "print(f\"x.y = {dot(x,y)}\")"
   ]
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     "text": [
      "||x|| = [[4.69041576]], ||y|| = [[3.]]\n"
     ]
    }
   ],
   "source": [
    "print(f\"||x|| = {norm(x)}, ||y|| = {norm(y)}\")"
   ]
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   "source": [
    "#### Angle between two vectors"
   ]
  },
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   "source": [
    "def angle(x,y):\n",
    "    if norm(x) == 0 or norm(y) == 0:\n",
    "        print(\"Invalid input: One of the vectors is zero.\")\n",
    "        return\n",
    "    theta_rad = np.acos(dot(x,y)/(norm(x)*norm(y)))\n",
    "    theta_deg = theta_rad*180/np.pi\n",
    "    return theta_rad, theta_deg"
   ]
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     "text": [
      "The angle between x and y is [[1.57079633]] radian, which is [[90.]] degrees.\n"
     ]
    }
   ],
   "source": [
    "theta_rad, theta_deg = angle(x,y)\n",
    "print(f\"The angle between x and y is {theta_rad} radian, which is {theta_deg} degrees.\")"
   ]
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   "source": [
    "----"
   ]
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   "source": [
    "## Projections\n",
    "\n",
    "Recall that the projection of a vector $x \\in \\mathbb{R}^{n}$ onto another vector $y \\in \\mathbb{R}^{n}$ is\n",
    "\n",
    "$\\mathrm{proj}_{y}(x) = \\left(\\dfrac{x \\cdot y}{\\| y \\|^{2}}\\right) y.$\n",
    "\n",
    "Let us write a function for this operator."
   ]
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   "source": [
    "def proj(x,y):\n",
    "    \"\"\" Projection of x onto y\"\"\"\n",
    "    p = dot(x,y)/(norm(y)**2)*y\n",
    "    return p"
   ]
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     "text": [
      "[[0.5]\n",
      " [0.5]]\n"
     ]
    }
   ],
   "source": [
    "x = np.array([[1, 0]]).T\n",
    "y = np.array([[1,1]]).T\n",
    "print(proj(x,y))"
   ]
  },
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