Linear algebra is the mathematical foundation upon which many modern fields stand — from computer graphics and cryptography to machine learning and AI. However, due to the dense nature of its symbols, theorems, and concepts, mastering linear algebra can feel daunting. This is where a Linear Algebra Cheat Sheet becomes your best ally.
Whether you’re a student trying to pass your finals or a machine learning engineer navigating high-dimensional vector spaces, a cheat sheet gives you instant access to crucial definitions, formulas, and computational tricks — all in one visual-friendly reference.
In this guide, we’ll explore:
- What makes a great linear algebra cheat sheet
- Popular downloadable PDFs
- Example-based references
- ML-focused cheat sheets
- Recommendations and usage tips
Let’s jump into the math matrix!
📌 Why You Need a Linear Algebra Cheat Sheet
Linear algebra concepts frequently resurface in:
- Matrix manipulations for deep learning
- Solving systems of equations in engineering
- Dimensionality reduction with PCA in data science
- Graphics rendering pipelines
Cheat sheets serve as compact study aids that cut through textbook clutter. Instead of flipping through pages of derivations, you can:
- Recall definitions quickly (e.g., span, null space, orthonormality)
- Apply algorithms like LU decomposition or least squares in seconds
- Check identities such as (AB)T = BTAT
🧮 Core Topics Every Cheat Sheet Should Include
A well-designed cheat sheet must cover these essentials:
1. Vector Basics
- Scalar vs. vector vs. matrix
- Dot and cross product
- Vector norms (L1, L2)
2. Matrix Operations
- Addition, subtraction
- Multiplication rules
- Transpose, inverse
- Rank and trace
3. Special Matrices
- Diagonal, identity, symmetric, orthogonal
- Sparse and dense matrices
4. System of Equations
- Augmented matrices
- Gaussian elimination
- Homogeneous and non-homogeneous solutions
5. Eigenvalues and Eigenvectors
- Characteristic polynomial
- Diagonalization
- SVD (Singular Value Decomposition)
6. Linear Transformations
- Mapping from one space to another
- Kernel and image
- Rank-nullity theorem
7. Applications
- Least Squares Approximation
- Principal Component Analysis (PCA)
- Projections onto subspaces
📄 Top Linear Algebra Cheat Sheets (with PDF links)
Here are the most trusted, expert-level cheat sheets available online — ideal for students, professionals, and researchers.
1. Laurent Lessard’s Linear Algebra Review
- Designed for ECE and control systems students
- Includes examples on least-squares problems and orthogonality
- Simple, diagram-aided layout
2. MIT 18.085 Linear Algebra Sheet
- Concise but deeply informative
- Shows properties of square vs. rectangular matrices
- Useful for proof-based questions
3. Yiping Lu’s ML-Focused Sheet
- Tailored for machine learning
- Features projection theory, Gram-Schmidt, SVD
- Aesthetic and clean layout
4. Ji‑Ha Kim Linear Algebra for ML (2025)
- Ideal for 2025 learners focusing on ML
- Includes step-by-step examples on QR, LU decompositions
- Formula-rich, ML-friendly notation
📥 [Available via Stanford ML Resources]
5. Minireference’s “Linear Algebra in 4 Pages”
- Old but gold — covers vector spaces, matrices, Gaussian elimination
- Lightweight yet powerful
📥 Download
6. Cheatography by Gustavhelms
- Community-sourced, regularly updated
- Contains row/column operations, null space, eigenvalues
- Great for high school to undergrad level
🔍 Linear Algebra Cheat Sheet with Examples
Examples bring clarity to abstract notation. Below are a few example-infused snippets pulled from the best sheets:
Example 1: Solving Linear System
Solve:
\[ \begin{cases} x + 2y = 4 \\ 3x + 4y = 10 \end{cases} \]
Matrix Form:
\[ AX = B \Rightarrow \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ 10 \end{bmatrix} \]
Solution:
Using row reduction or matrix inversion:
\[ X = A^{-1}B \]
Example 2: Orthogonal Projection
Project b onto subspace A:
\[ \hat{x} = (A^\top A)^{-1} A^\top b \]
Use case: In least squares for regression, especially in ML.
🤖 Cheat Sheets for Machine Learning
If you’re working with ML frameworks like TensorFlow, PyTorch, or building algorithms from scratch, linear algebra is unavoidable.
ML Topics Covered in Cheat Sheets:
- Dot products for similarity
- Matrix multiplications in feed-forward layers
- Orthogonalization for regularization
- SVD in collaborative filtering (recommender systems)
Best ML Cheat Sheets:
- Yiping Lu
- Ji-Ha Kim
- MIT + PCA-focused supplements
If you’re struggling with understanding how math powers ML, these cheat sheets bridge theory and practice.
📚 How to Use Cheat Sheets Effectively
- Print & Pin: Stick one near your desk or whiteboard
- Use as Flashcards: Break it into sections for spaced learning
- Link to Real Problems: Use cheat sheets when debugging ML code or solving system equations
- Cross-reference Examples: Don’t memorize, apply with exercises
📝 Final Thoughts on the Best Linear Algebra Cheat Sheets
Linear algebra is the heartbeat of modern computational science. From understanding neural networks to transforming 3D models, it’s everywhere. But it doesn’t need to be hard. With a well-structured cheat sheet, your grasp of vectors, matrices, and transformations becomes stronger and more intuitive.
The key is to choose the cheat sheet that fits your learning style:
- Want depth? → MIT or Ji‑Ha Kim.
- Prefer visuals? → Yiping Lu or Cheatography.
- Need application? → Laurent Lessard or Minireference.
You now have a complete toolkit to download, review, and master linear algebra like a pro.
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