diff --git a/HW1.ipynb b/HW1.ipynb
index dba7747..cabcbb3 100644
--- a/HW1.ipynb
+++ b/HW1.ipynb
@@ -1,132 +1,144 @@
 {
  "cells": [
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "# Homework \\#1 - Python exercise"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "import numpy as np"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "  Consider the following change of basis matrix in $\\mathbb{C}^8$, with respect to the standard orthonormal basis:\n",
     "  \n",
     "  $$\n",
     "      \\mathbf{H} = \\begin{bmatrix}\n",
     "  1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
     "  0 & 0 & 1 & -1 & 0 & 0 & 0 & 0\\\\\n",
     "  0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\\\\n",
     "  0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\\\\n",
     "  1 & 1 & -1 & -1 & 0 & 0 & 0 & 0\\\\\n",
     "  0 & 0 & 0 & 0 & 1 & 1 & -1 & -1\\\\\n",
     "  1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\\\\n",
     "  1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n",
     "  \\end{bmatrix}.\n",
     "  $$\n",
     "  \n",
     "* What is an easy way to prove that the rows in $\\mathbf{H}$ do indeed form a basis? (\\emph{Hint}: it's enough to show that they are linearly independent, which is to say, that the matrix has full rank...)\n",
     "* Use Python to verify point (a); obviously you can use \\texttt{numpy}.\n",
     "\n",
     "\n",
     "The basis described by $\\mathbf{H}$ is called the \\emph{Haar basis} and it is one of the most celebrated cornerstones of a branch of signal processing called wavelet analysis (which we won't study in this class). To get a feeling for its properties, however, consider the following set of Python experiments:\n",
     "\n",
     "* Verify that $\\mathbf{H}\\mathbf{H}^H$ is a diagonal matrix, which means the vectors are orthogonal.\n",
     "* Consider a constant signal $\\mathbf{x} = \\begin{bmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\end{bmatrix}$  and compute its coefficients in the Haar basis.\n",
     "* Consider an alternating signal $\\mathbf{y} = \\begin{bmatrix}1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\end{bmatrix}$ and compute its coefficients in the Haar basis."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "# 8x8 Haar matrix\n",
     "H = np.array([\n",
     "    [1, -1,  0,  0,  0,  0,  0,  0],\n",
     "    [0,  0,  1, -1,  0,  0,  0,  0],\n",
     "    [0,  0,  0,  0,  1, -1,  0,  0],\n",
     "    [0,  0,  0,  0,  0,  0,  1, -1],\n",
     "    [1,  1, -1, -1,  0,  0,  0,  0],\n",
     "    [0,  0,  0,  0,  1,  1, -1, -1],\n",
     "    [1,  1,  1,  1, -1, -1, -1, -1],\n",
     "    [1,  1,  1,  1,  1,  1,  1,  1]\n",
     "])"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "# b) show that the rows are linearly independent\n",
     "# ..."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
-    "# c) verify that HH^T is symmetric\n",
+    "# c) verify that HH^T is diagonal\n",
     "# ..."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "# d) compute a basis expansion\n",
     "x = np.ones((8,1)) # note that we specify the shape to obtain a column vector\n",
     "\n",
     "# ..."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "# e) compute a basis expansion\n",
     "y = ..."
    ]
   }
  ],
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    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.7"
+   "version": "3.6.3"
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 }