diff --git a/lecture_material/21-linalg-2/21-linalg2.ipynb b/lecture_material/21-linalg-2/21-linalg2.ipynb
index 2b412c581c17b6eb1a9160d019be7b308e01251e..82020b73044d88051004bdc787d2e150133408af 100644
--- a/lecture_material/21-linalg-2/21-linalg2.ipynb
+++ b/lecture_material/21-linalg-2/21-linalg2.ipynb
@@ -1788,7 +1788,7 @@
     "Define <font color=\"red\">$P = X(X^TX)^{-1}X^T$</font>.  This is a **projection matrix**.  If you multiply a vector by $P$, you get back a new vector of the same size, with two properties:\n",
     "\n",
     "1. it will be in the column space of $X$\n",
-    "2. the new vector will be as \"close as possible\" to the original vector\n",
+    "2. the new vector will be as \"close\" as possible to the original vector\n",
     "\n",
     "Note: computing P is generally very expensive.\n",
     "\n",
@@ -2053,7 +2053,7 @@
    "id": "1d8e442c",
    "metadata": {},
    "source": [
-    "The new vector will be as \"close as possible\" to the original vector."
+    "The new vector will be as \"close\" as possible to the original vector."
    ]
   },
   {
diff --git a/lecture_material/21-linalg-2/21-linalg2_001.ipynb b/lecture_material/21-linalg-2/21-linalg2_001.ipynb
index 888f8f29e3671fe78245d31d13fb7ff39b6f850c..438070f8e02fa0e96119c6711288009562dbaf0b 100644
--- a/lecture_material/21-linalg-2/21-linalg2_001.ipynb
+++ b/lecture_material/21-linalg-2/21-linalg2_001.ipynb
@@ -1048,7 +1048,7 @@
     "Define <font color=\"red\">$P = X(X^TX)^{-1}X^T$</font>.  This is a **projection matrix**.  If you multiply a vector by $P$, you get back another vector of the same size, with two properties:\n",
     "\n",
     "1. it will be in the column space of $X$\n",
-    "2. the new vector will be as \"close as possible\" to the original vector\n",
+    "2. the new vector will be as \"close\" as possible to the original vector\n",
     "\n",
     "Note: computing P is generally very expensive.\n",
     "\n",
@@ -1124,7 +1124,7 @@
    "id": "1d8e442c",
    "metadata": {},
    "source": [
-    "The new vector will be as \"close as possible\" to the original vector."
+    "The new vector will be as \"close\" as possible to the original vector."
    ]
   },
   {
diff --git a/lecture_material/21-linalg-2/21-linalg2_002.ipynb b/lecture_material/21-linalg-2/21-linalg2_002.ipynb
index 888f8f29e3671fe78245d31d13fb7ff39b6f850c..438070f8e02fa0e96119c6711288009562dbaf0b 100644
--- a/lecture_material/21-linalg-2/21-linalg2_002.ipynb
+++ b/lecture_material/21-linalg-2/21-linalg2_002.ipynb
@@ -1048,7 +1048,7 @@
     "Define <font color=\"red\">$P = X(X^TX)^{-1}X^T$</font>.  This is a **projection matrix**.  If you multiply a vector by $P$, you get back another vector of the same size, with two properties:\n",
     "\n",
     "1. it will be in the column space of $X$\n",
-    "2. the new vector will be as \"close as possible\" to the original vector\n",
+    "2. the new vector will be as \"close\" as possible to the original vector\n",
     "\n",
     "Note: computing P is generally very expensive.\n",
     "\n",
@@ -1124,7 +1124,7 @@
    "id": "1d8e442c",
    "metadata": {},
    "source": [
-    "The new vector will be as \"close as possible\" to the original vector."
+    "The new vector will be as \"close\" as possible to the original vector."
    ]
   },
   {
diff --git a/lecture_material/22-classification/Confusion_matrix_binary.png b/lecture_material/22-classification/Confusion_matrix_binary.png
index 8f4856637550b2d48da766a22ea7202999ea9127..3f91eb0da9ad50bdeb75cd3f8d070ac3e4dd9c6d 100644
Binary files a/lecture_material/22-classification/Confusion_matrix_binary.png and b/lecture_material/22-classification/Confusion_matrix_binary.png differ