diff --git a/AGENTS.md b/AGENTS.md
index ecb75ec..f279cbc 100644
--- a/AGENTS.md
+++ b/AGENTS.md
@@ -123,6 +123,22 @@ myst build --html
 - `--execute` runs all code cells and caches results in `_build/execute/`
 - `--html` produces HTML output in `_build/html/`
 
+### Keeping the build warning-free
+
+The CI `build` job runs the real build, so keep it free of warnings. The three
+warning classes seen so far (all fixed in #11):
+
+- **`missing heading depth N`** — a page jumps heading levels (e.g. `#` then
+  `###`, skipping `##`). Use consecutive depths.
+- **`textEnv, Too few columns specified in the {array} column argument`** — a
+  `\begin{array}{...}` declares fewer columns than a row has. Match the spec to
+  the actual column count (e.g. a 4×4 matrix needs `{cccc}`, not `{ccc}`).
+- **`Language is not defined for code block`** — a `{code-cell}` with no
+  language. Add `python` (``{code-cell} python``). Note this only surfaces for
+  lessons referenced in `myst.yml` as **`.md`** (currently lesson 0): lessons
+  1–5 are referenced as **`.ipynb`**, where Jupyter carries the Python language,
+  so their bare `{code-cell}` directives are fine.
+
 ### CI/CD
 
 The GitHub Actions workflow (`.github/workflows/notebook_to_html.yml`) runs on pushes to `main` and on pull requests:
diff --git a/basic_lessons/README.md b/basic_lessons/README.md
index 5dc7aa4..feab50e 100644
--- a/basic_lessons/README.md
+++ b/basic_lessons/README.md
@@ -4,13 +4,13 @@ In this six-lesson tutorial, we start from the very basics of setting up your Py
 then cover scalar and matricial operations in Python using `numpy`,
 all the way until the basics of kinematic control. Until kinematic control, most is based on [@spong2020robot].
 
-# Using this book
+## Using this book
 
 Each lesson is a [MyST text notebook](https://mystmd.org/guide/notebooks-with-markdown). Each lesson can be
 opened and executed with popular IDEs, such as [VSCode](https://code.visualstudio.com) and [PyCharm](https://www.jetbrains.com/pycharm/).
 The reader is expected to follow it sequentially.
 
-# Contents
+## Contents
 
 
 | Number | Title and Link               | Content                                                                                                                                                                                                                                    |
diff --git a/basic_lessons/lesson0_tutorial.md b/basic_lessons/lesson0_tutorial.md
index 63d804f..97ff198 100644
--- a/basic_lessons/lesson0_tutorial.md
+++ b/basic_lessons/lesson0_tutorial.md
@@ -71,7 +71,7 @@ pip install numpy matplotlib
 
 You can verify that the packages are correctly installed by running the cell below.
 
-````{code-cell}
+````{code-cell} python
 import numpy as np
 import matplotlib
 import matplotlib.pyplot as plt
diff --git a/basic_lessons/lesson2_tutorial.md b/basic_lessons/lesson2_tutorial.md
index cfdf0f4..d24c4ed 100644
--- a/basic_lessons/lesson2_tutorial.md
+++ b/basic_lessons/lesson2_tutorial.md
@@ -322,7 +322,7 @@ We can also perform pose transformations using sequential right multiplications
 
 For example, consider a translation in 3D along the _World_ frame, represented by the homogeneous transformation matrix below.
 
-$$\mymatrix{H}^0_a = \mymatrix{H}_a = \left[\begin{array}{ccc}
+$$\mymatrix{H}^0_a = \mymatrix{H}_a = \left[\begin{array}{cccc}
                                          1 & 0 & 0 & 1 \\
                                          0 & 1 & 0 & 2 \\
                                          0 & 0 & 1 & 3 \\
@@ -331,7 +331,7 @@ $$\mymatrix{H}^0_a = \mymatrix{H}_a = \left[\begin{array}{ccc}
 
 Consider a rotation in 3D about the _current_ frame, represented by the homogeneous transformation matrix below.
 
-$$\mymatrix{H}^a_b = \left[\begin{array}{ccc}
+$$\mymatrix{H}^a_b = \left[\begin{array}{cccc}
                      \cos{\theta_{ab}} & -\sin{\theta_{ab}} & 0 & 0 \\
                      \sin{\theta_{ab}} & \cos{\theta_{ab}}  & 0 & 0 \\
                      0                 & 0                  & 1 & 0 \\