MML Exercise Answers¶
Worked solutions to the end-of-chapter exercises in Mathematics for Machine Learning (Deisenroth, Faisal & Ong). The full book is freely available here.
Each question is laid out the same way: a Topics & Definitions box, a plain-English explanation, the worked steps (with real matrices), then the answer. New to the notation? See the list of symbols at the bottom of this page.
Finding this useful?
Star the repo on GitHub to follow along, and if you spot a mistake, open an issue so I can fix it.
Progress¶
| Chapter | Exercises answered |
|---|---|
| 2 · Linear Algebra | ✅ 2.1–2.20 complete |
| 3 · Analytic Geometry | ✅ 3.1–3.10 complete |
| 4 · Matrix Decompositions | ✅ 4.1–4.12 complete |
| 5 · Vector Calculus | ✅ 5.1–5.9 complete + 5.10 bonus |
| 6 · Probability & Distributions | ✅ 6.1, 6.2a |
| 7 · Continuous Optimization | ⬜ |
| 8 · When Models Meet Data | ⬜ |
| 9 · Linear Regression | ⬜ |
| 10 · Dimensionality Reduction (PCA) | ⬜ |
| 11 · Density Estimation (GMM) | ⬜ |
| 12 · Classification (SVM) | ⬜ |
Helpful resources¶
Stuck reducing a matrix to RREF?
This calculator shows every elimination step, which is great for checking your own work: emathhelp RREF calculator
For building real intuition on vectors, transformations, and matrices, 3Blue1Brown's Essence of Linear Algebra video series is fantastic and highly recommended.
For linear mappings specifically, this video is a clear, helpful walkthrough.
Another helpful video worth a watch.
For the calculus side of things (Chapter 5 onward), Paul's Online Math Notes are a thorough, well-explained reference for Calculus I, II, and III.
How this site is built¶
Static site generated with MkDocs + Material, math rendered with KaTeX.
See the README.md in the repo for how to run it locally and deploy to GitHub Pages.
Conventions
- Pastel aesthetic (lilac / sage / pink)
- Each question: definitions box → explanation → worked steps → answer
- Math written in LaTeX, rendered by KaTeX
List of symbols¶
A quick reference for the notation used across these answers. Skim it whenever a symbol looks unfamiliar.
Sets and spaces¶
| Symbol | Meaning |
|---|---|
| \(\mathbb{R}\) | the real numbers |
| \(\mathbb{R}^n\) | the space of real vectors with \(n\) entries |
| \(\mathbb{Z}_n\) | the integers modulo \(n\): \(\{[0],[1],\dots,[n-1]\}\) |
| \(\in\) | "is an element of" |
| \(\subseteq\) | "is a subset of" |
| \(\{\,\cdot\mid\cdot\,\}\) | set-builder: \(\{\,x \mid \text{condition}\,\}\) |
| \(\varnothing\) | the empty set (no solutions) |
Vectors and matrices¶
| Symbol | Meaning |
|---|---|
| \(\mathbf{x},\ \mathbf{y}\) | vectors (bold lowercase) |
| \(\mathbf{0}\) | the zero vector |
| \(x_i\) | the \(i\)-th entry of \(\mathbf{x}\) |
| \(A,\ B\) | matrices (uppercase) |
| \(A^\top,\ \mathbf{x}^\top\) | transpose (rows become columns) |
| \(A^{-1}\) | matrix inverse (\(AA^{-1} = I\)) |
| \(I\) | the identity matrix |
| \(\det A\) | determinant of \(A\) |
| \(\operatorname{rank}(A),\ \operatorname{rk}(A)\) | rank: the number of pivots |
Operations¶
| Symbol | Meaning |
|---|---|
| \(\mathbf{x}^\top\mathbf{y}\) | dot product (standard inner product) |
| \(\langle\mathbf{x},\mathbf{y}\rangle\) | inner product (possibly weighted, \(\mathbf{x}^\top A\mathbf{y}\)) |
| \(\lVert\mathbf{x}\rVert\) | norm (length): \(\sqrt{\langle\mathbf{x},\mathbf{x}\rangle}\) |
| \(\star,\ \oplus,\ \otimes\) | custom operations defined within a problem |
| \(\circ\) | composition of maps: \((g\circ f)(x) = g(f(x))\) |
| \(\approx\) | approximately equal |
| \(\neq\) | not equal |
Maps and subspaces¶
| Symbol | Meaning |
|---|---|
| \(\Phi,\ f,\ g\) | linear maps (homomorphisms) |
| \(\mapsto\) | "maps to": \(\mathbf{x}\mapsto\Phi(\mathbf{x})\) |
| \(A_\Phi\) | the transformation matrix of \(\Phi\) |
| \(\ker(\Phi)\) | kernel: inputs sent to \(\mathbf{0}\) |
| \(\operatorname{Im}(\Phi)\) | image: all reachable outputs |
| \(\dim(U)\) | dimension of a subspace |
| \(\operatorname{span}[\cdots]\) | span: all linear combinations of the listed vectors |
| \(U_1 \cap U_2\) | intersection: vectors lying in both subspaces |
| \(U_1 + U_2\) | sum: all \(\mathbf{u}_1 + \mathbf{u}_2\) |
| \(\lambda\) | a scalar coordinate or an eigenvalue |
Geometry¶
| Symbol | Meaning |
|---|---|
| \(d(\mathbf{x},\mathbf{y})\) | distance: \(\lVert\mathbf{x}-\mathbf{y}\rVert\) |
| \(\theta\) | angle between two vectors |
| \(\pi_U(\mathbf{x})\) | orthogonal projection of \(\mathbf{x}\) onto the subspace \(U\) |