diff --git a/pct/modules/04/linear-equations.ipynb b/pct/modules/04/linear-equations.ipynb
index 294a3e2..48d043d 100644
--- a/pct/modules/04/linear-equations.ipynb
+++ b/pct/modules/04/linear-equations.ipynb
@@ -69,7 +69,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -106,7 +106,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [
     {
@@ -144,21 +144,26 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Non-Inversion Method\n",
-    "\n",
-    "\n",
-    "But it is never a good idea to use the inverse of a matrix to solve the linear equations, because it can be \n",
+    "But it is **never a good idea** to use matrix inverse to solve the linear equations, because it can be\n",
     "- numerically unstable\n",
     "- computationally expensive\n",
     "\n",
-    "For interested readers, here's a good writeup on [why one should never invert a matrix for Ax=b](https://gregorygundersen.com/blog/2020/12/09/matrix-inversion/).\n",
+    "For interested readers, here's a good writeup on [why one should never invert a matrix for Ax=b](https://gregorygundersen.com/blog/2020/12/09/matrix-inversion/).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Non-Inversion Method\n",
+    "\n",
     "\n",
-    "In Python, we can use the `np.linalg.solve` function to solve the linear equations.\n"
+    "Instead, we can use the `np.linalg.solve` function to solve the linear equations.\n"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [
     {
@@ -195,7 +200,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
@@ -251,7 +256,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
@@ -290,7 +295,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
@@ -362,7 +367,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -394,7 +399,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -441,7 +446,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
@@ -511,7 +516,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -545,14 +550,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "File downloaded and saved to /tmp/tmpeno7e_rq.m\n",
+      "File downloaded and saved to /tmp/tmpgjvea826.m\n",
       "File size: 1530051 bytes\n"
      ]
     }
@@ -593,7 +598,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -621,7 +626,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
@@ -631,7 +636,7 @@
        "\twith 154958 stored elements and shape (19928, 19928)>"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 13,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -642,14 +647,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "27.2 ms ± 594 μs per loop (mean ± std. dev. of 5 runs, 20 loops each)\n"
+      "27 ms ± 560 μs per loop (mean ± std. dev. of 5 runs, 20 loops each)\n"
      ]
     }
    ],
@@ -674,14 +679,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 15,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "40.2 ms ± 3.57 ms per loop (mean ± std. dev. of 5 runs, 20 loops each)\n"
+      "36.6 ms ± 1.13 ms per loop (mean ± std. dev. of 5 runs, 20 loops each)\n"
      ]
     }
    ],
@@ -747,7 +752,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -769,14 +774,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "41.8 ms ± 3.52 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
+      "37.4 ms ± 578 μs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
      ]
     }
    ],
@@ -799,14 +804,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "10.2 ms ± 52.4 μs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
+      "10 ms ± 57.9 μs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
      ]
     }
    ],
@@ -830,6 +835,15 @@
     "\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In addition, sparse LU factorization results can be reused for different\n",
+    "right-hand sides, just like in the dense canse. This will be discussed in\n",
+    "the section for acceleration techniques."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},