2.1 Scalars Vectors Matrices and Tensors.ipynb

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    "# Introduction\n",
    "\n",
    "The basic elements used in linear algebra and their definitions. It also introduces important functions in Python/Numpy that we will use all along this series. It will explain how to create and use vectors and matrices through examples."
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   "source": [
    "import numpy as np   # N-Dim Array object \n",
    "import matplotlib.pyplot as plt  # Data Visulization\n",
    "import seaborn as sns  # Data Visulization"
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    "# 2.1 Scalars, Vectors, Matrices and Tensors\n",
    "\n",
    "Let's start with some basic definitions:\n",
    "\n",
    "<img src=\"images/scalar-vector-matrix-tensor.png\" width=\"1000\" alt=\"An example of a scalar, a vector, a matrix and a tensor\" title=\"Difference between a scalar, a vector, a matrix and a tensor\">\n",
    "\n",
    "<em>Difference between a scalar, a vector, a matrix and a tensor</em>\n",
    "\n",
    "- A scalar is a single number\n",
    "- A Scalar is has only magnitude (size)\n",
    "\n",
    "**Eg:**\n",
    " 3.044, −7 and 2½ are scalars\n",
    " \n",
    " - Distance, speed, time, temperature, mass, length, area, volume, density, charge, pressure, energy, work and power are all scalars.\n",
    " \n",
    " "
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   "source": [
    "Scalar = np.array(-7)\n",
    "Scalar"
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    "**ndarray.ndim** the number of axes (dimensions) of the array."
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    "Scalar.ndim  # Scalar Don't have a dimensions "
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    "**ndarray.dtype**\n",
    "* An object describing the type of the elements in the array. \n",
    "* One can create or specify dtype’s using standard Python types. \n",
    "* Additionally NumPy provides types of its own. \n",
    "    * numpy.int32, \n",
    "    * numpy.int16, and \n",
    "    * numpy.float64 are some examples."
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    "**ndarray.shape**\n",
    "* The dimensions of the array.\n",
    "* This is a tuple of integers indicating the size of the array in each dimension. \n",
    "* For a matrix with `n rows` and `m columns`, shape will be `(n,m)`. \n",
    "* The length of the shape tuple is therefore the number of axes, ndim."
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    "**ndarray.size**\n",
    "* To identify total number of elements in an array.\n",
    "* his is equal to the product of the elements of shape."
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    "**ndarray.itemsize**\n",
    "\n",
    "* The size in bytes of each element of the array. \n",
    "* For example, an array of elements of type float64 has itemsize 8 (=64/8), while one of type complex32 has itemsize 4 (=32/8). \n",
    "\n",
    "* It is equivalent to ndarray.dtype.itemsize.\n",
    "\n",
    "\n",
    "**Eg:** \n",
    "\n",
    "np.array([1.5,2.5,3.5,4.5,5.5,6.5,7.7,8.5],dtype=np.float64)\n"
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    "Scalar.itemsize  # Memory which is allocated to the each Element in an array"
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    "**ndarray.data**\n",
    "* The buffer containing the actual elements of the array.\n",
    "* Normally, we won’t need to use this attribute because we will access the elements in an array using indexing facilities."
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    "### Vector\n",
    "- A vector is an array of numbers.\n",
    "- A vector has magnitude and direction:\n",
    "\n",
    "[]\n",
    "\n",
    "$$\n",
    "\\bs{x} =\\begin{bmatrix}\n",
    "    x_1 \\\\\\\\\n",
    "    x_2 \\\\\\\\\n",
    "    \\cdots \\\\\\\\\n",
    "    x_n\n",
    "\\end{bmatrix}\n",
    "$$\n"
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    "### Example 1: Create a vector with Python and Numpy\n",
    "\n",
    "\n",
    "![](https://lh3.googleusercontent.com/-wSyXTYTFR6Y/Xyave8XRo-I/AAAAAAAApf0/8jV9cOyJjJYPL440nX0jU7QrN3Hy5rWtwCK8BGAsYHg/s0/2020-08-02.png)"
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x)\n",
    "plt.xlabel(\"X-Axis\")\n",
    "plt.ylabel(\"Y-Axis\")\n",
    "plt.title(\"Vector\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nteract": {
     "transient": {
      "deleting": false
     }
    }
   },
   "source": [
    "**Examples for Vectors:**Displacement, velocity, acceleration, force and momentum are all vectors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nteract": {
     "transient": {
      "deleting": false
     }
    }
   },
   "source": [
    "###### Matrix\n",
    "- A matrix is a 2-D array\n",
    "\n",
    "$$\n",
    "\\bs{A}=\n",
    "\\begin{bmatrix}\n",
    "    A_{1,1} & A_{1,2} & \\cdots & A_{1,n} \\\\\\\\\n",
    "    A_{2,1} & A_{2,2} & \\cdots & A_{2,n} \\\\\\\\\n",
    "    \\cdots & \\cdots & \\cdots & \\cdots \\\\\\\\\n",
    "    A_{m,1} & A_{m,2} & \\cdots & A_{m,n}\n",
    "\\end{bmatrix}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 2.\n",
    "\n",
    "#### Create a (3x2) matrix with nested brackets\n",
    "\n",
    "The `array()` function can also create $2$-dimensional arrays with nested brackets:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0],\n",
       "       [3, 4],\n",
       "       [5, 6]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[1,0], \n",
    "              [3, 4], \n",
    "              [5, 6]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "nteract": {
     "transient": {
      "deleting": false
     }
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 2)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "nteract": {
     "transient": {
      "deleting": false
     }
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.ndim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "nteract": {
     "transient": {
      "deleting": false
     }
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(dtype('int32'), 4, <memory at 0x000002338F6F72D0>)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dtype,A.itemsize,A.data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "nteract": {
     "transient": {
      "deleting": false
     }
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 0],\n",
       "        [3, 4],\n",
       "        [5, 6]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Mx = np.matrix([[1,0], \n",
    "              [3, 4], \n",
    "              [5, 6]])\n",
    "Mx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2338f77a668>,\n",
       " <matplotlib.lines.Line2D at 0x2338f77a6d8>]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 1, 0],\n",
       "       [2, 6, 4, 1],\n",
       "       [5, 2, 8, 6],\n",
       "       [3, 2, 5, 1]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e=np.array([[1,0,1,0],[2,6,4,1],[5,2,8,6],[3,2,5,1]])\n",
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2338f7e9f28>,\n",
       " <matplotlib.lines.Line2D at 0x2338f7e9f98>,\n",
       " <matplotlib.lines.Line2D at 0x2338f7f7080>,\n",
       " <matplotlib.lines.Line2D at 0x2338f7f7128>]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-08-02T12:45:22.280Z",
     "iopub.status.busy": "2020-08-02T12:45:22.243Z",
     "iopub.status.idle": "2020-08-02T12:45:22.840Z",
     "shell.execute_reply": "2020-08-02T12:45:22.896Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x2156a8e03c8>"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x396 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.gray()\n",
    "plt.imshow(e) # Converting array to image"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([array([0., 1., 1., 1., 0., 0., 1., 0., 0., 0.]),\n",
       "  array([1., 0., 2., 0., 0., 0., 0., 1., 0., 0.]),\n",
       "  array([0., 1., 0., 0., 0., 1., 1., 0., 0., 1.]),\n",
       "  array([1., 2., 0., 0., 0., 0., 0., 1., 0., 0.])],\n",
       " array([0. , 0.8, 1.6, 2.4, 3.2, 4. , 4.8, 5.6, 6.4, 7.2, 8. ]),\n",
       " <a list of 4 Lists of Patches objects>)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(e) # Data Distribution "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Shape\n",
    "\n",
    "The shape of an array (that is to say its dimensions) tells you the number of values for each dimension. For a $2$-dimensional array it will give you the number of rows and the number of columns. Let's find the shape of our preceding $2$-dimensional array `A`. Since `A` is a Numpy array (it was created with the `array()` function) you can access its shape with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-08-02T12:47:19.570Z",
     "iopub.status.busy": "2020-08-02T12:47:19.525Z",
     "iopub.status.idle": "2020-08-02T12:47:19.647Z",
     "shell.execute_reply": "2020-08-02T12:47:19.689Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 2)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that $\\bs{A}$ has 3 rows and 2 columns.\n",
    "\n",
    "Let's check the shape of our first vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-08-02T12:47:29.102Z",
     "iopub.status.busy": "2020-08-02T12:47:29.058Z",
     "iopub.status.idle": "2020-08-02T12:47:29.208Z",
     "shell.execute_reply": "2020-08-02T12:47:29.264Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4,)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, you can see that $\\bs{x}$ has only one dimension. The number corresponds to the length of the array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-08-02T12:47:34.860Z",
     "iopub.status.busy": "2020-08-02T12:47:34.824Z",
     "iopub.status.idle": "2020-08-02T12:47:34.990Z",
     "shell.execute_reply": "2020-08-02T12:47:35.054Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Transposition\n",
    "\n",
    "With transposition you can convert a row vector to a column vector and vice versa:\n",
    "\n",
    "<img src=\"images/vector-transposition.png\" alt=\"Transposition of a vector\" title=\"Vector transposition\" width=\"200\">\n",
    "<em>Vector transposition</em>\n",
    "\n",
    "The transpose $\\bs{A}^{\\text{T}}$ of the matrix $\\bs{A}$ corresponds to the mirrored axes. If the matrix is a square matrix (same number of columns and rows):\n",
    "\n",
    "<img src=\"images/square-matrix-transposition.png\" alt=\"Transposition of a square matrix\" title=\"Square matrix transposition\" width=\"300\">\n",
    "<em>Square matrix transposition</em>\n",
    "\n",
    "If the matrix is not square the idea is the same:\n",
    "\n",
    "<img src=\"images/non-squared-matrix-transposition.png\" alt=\"Transposition of a square matrix\" title=\"Non square matrix transposition\" width=\"300\">\n",
    "<em>Non-square matrix transposition</em>\n",
    "\n",
    "\n",
    "The superscript $^\\text{T}$ is used for transposed matrices.\n",
    "\n",
    "$$\n",
    "\\bs{A}=\n",
    "\\begin{bmatrix}\n",
    "    A_{1,1} & A_{1,2} \\\\\\\\\n",
    "    A_{2,1} & A_{2,2} \\\\\\\\\n",
    "    A_{3,1} & A_{3,2}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\bs{A}^{\\text{T}}=\n",
    "\\begin{bmatrix}\n",
    "    A_{1,1} & A_{2,1} & A_{3,1} \\\\\\\\\n",
    "    A_{1,2} & A_{2,2} & A_{3,2}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "The shape ($m \\times n$) is inverted and becomes ($n \\times m$).\n",
    "\n",
    "<img src=\"images/dimensions-transposition-matrix.png\" alt=\"Dimensions of matrix transposition\" title=\"Dimensions of matrix transposition\" width=\"300\">\n",
    "<em>Dimensions of matrix transposition</em>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3.\n",
    "\n",
    "#### Create a matrix A and transpose it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 2],\n",
       "        [3, 4],\n",
       "        [5, 6]])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.matrix([[1, 2], [3, 4], [5, 6]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 3, 5],\n",
       "        [2, 4, 6]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_t = A.T\n",
    "A_t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check the dimensions of the matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x2338f862c50>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_t.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 2)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that the number of columns becomes the number of rows with transposition and vice versa."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Addition\n",
    "\n",
    "<img src=\"images/matrix-addition.png\" alt=\"Addition of two matrices\" title=\"Addition of two matrices\" width=\"300\">\n",
    "<em>Addition of two matrices</em>\n",
    "\n",
    "Matrices can be added if they have the same shape:\n",
    "\n",
    "$$\\bs{A} + \\bs{B} = \\bs{C}$$\n",
    "\n",
    "Each cell of $\\bs{A}$ is added to the corresponding cell of $\\bs{B}$:\n",
    "\n",
    "$$\\bs{A}_{i,j} + \\bs{B}_{i,j} = \\bs{C}_{i,j}$$\n",
    "\n",
    "$i$ is the row index and $j$ the column index.\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "    A_{1,1} & A_{1,2} \\\\\\\\\n",
    "    A_{2,1} & A_{2,2} \\\\\\\\\n",
    "    A_{3,1} & A_{3,2}\n",
    "\\end{bmatrix}+\n",
    "\\begin{bmatrix}\n",
    "    B_{1,1} & B_{1,2} \\\\\\\\\n",
    "    B_{2,1} & B_{2,2} \\\\\\\\\n",
    "    B_{3,1} & B_{3,2}\n",
    "\\end{bmatrix}=\n",
    "\\begin{bmatrix}\n",
    "    A_{1,1} + B_{1,1} & A_{1,2} + B_{1,2} \\\\\\\\\n",
    "    A_{2,1} + B_{2,1} & A_{2,2} + B_{2,2} \\\\\\\\\n",
    "    A_{3,1} + B_{3,1} & A_{3,2} + B_{3,2}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "The shape of $\\bs{A}$, $\\bs{B}$ and $\\bs{C}$ are identical. Let's check that in an example:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 4.\n",
    "\n",
    "#### Create two matrices A and B and add them\n",
    "\n",
    "With Numpy you can add matrices just as you would add vectors or scalars."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4],\n",
       "       [5, 6]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[1, 2], [3, 4], [5, 6]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x2338f82a3c8>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 5],\n",
       "       [7, 4],\n",
       "       [4, 3]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = np.array([[2, 5], [7, 4], [4, 3]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x2338f92f240>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3,  7],\n",
       "       [10,  8],\n",
       "       [ 9,  9]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Add matrices A and B\n",
    "C = A + B\n",
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x2338f987cc0>"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is also possible to add a scalar to a matrix. This means adding this scalar to each cell of the matrix.\n",
    "\n",
    "$$\n",
    "\\alpha+ \\begin{bmatrix}\n",
    "    A_{1,1} & A_{1,2} \\\\\\\\\n",
    "    A_{2,1} & A_{2,2} \\\\\\\\\n",
    "    A_{3,1} & A_{3,2}\n",
    "\\end{bmatrix}=\n",
    "\\begin{bmatrix}\n",
    "    \\alpha + A_{1,1} & \\alpha + A_{1,2} \\\\\\\\\n",
    "    \\alpha + A_{2,1} & \\alpha + A_{2,2} \\\\\\\\\n",
    "    \\alpha + A_{3,1} & \\alpha + A_{3,2}\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 5.\n",
    "\n",
    "#### Add a scalar to a matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4],\n",
       "       [5, 6]])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5,  6],\n",
       "       [ 7,  8],\n",
       "       [ 9, 10]])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Exemple: Add 4 to the matrix A\n",
    "C = A+4\n",
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10, 20],\n",
       "       [25, 27],\n",
       "       [26, 58]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ar = np.array([[10,20],[25,27],[26,58]])\n",
    "Ar                      "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([10,20,30])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nteract": {
     "transient": {
      "deleting": false
     }
    }
   },
   "source": [
    "### Tensor\n",
    "- A tensor is a $n$-dimensional array with $n>2$\n",
    "\n",
    "We will follow the conventions used in the [Machine Learning](https://github.com/reddyprasade/Machine-Learning-with-Scikit-Learn-Python-3.x):\n",
    "\n",
    "- scalars are written in lowercase and italics. For instance: $n$\n",
    "- vectors are written in lowercase, italics and bold type. For instance: $\\bs{x}$\n",
    "- matrices are written in uppercase, italics and bold. For instance: $\\bs{X}$"
   ]
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