Linear Algebra

Linear algebra forms the backbone of modern machine learning systems, yet many candidates struggle with its practical implementation in coding interviews. This guide bridges the gap between theoretical understanding and production-ready implementation, offering a structured approach to mastering linear algebra concepts through coding challenges. You'll find a carefully curated progression of problems, common pitfalls identified from real interview experiences, and connections to real-world ML applications.

Core Knowledge

Linear Algebra Fundamentals

Numerical Stability

Matrix Factorization Techniques

Implementation Mastery

Key Questions

Status	Question	Category
	Diagonal matrix from vector (space-efficient)	Linear Algebra
	Matrix Inverse with Regularization	Linear Algebra
	Calculate Covariance Matrix	Linear Algebra
	Matrix Determinant Calculation	Linear Algebra
	Sparse Vector-Vector Multiplication	Linear Algebra
	Sparse Matrix-Vector Multiplication (CSR vs COO)	Linear Algebra
	Eigenvalues of a Square Matrix	Linear Algebra
	Transformation Matrix from Basis B to C	Linear Algebra
	Implement Orthogonal Projection of Vector onto Line	Linear Algebra
	QR Decomposition Implementation	Linear Algebra

Common Pitfalls

Extended Questions

Status	Question	Category
	Parallel Matrix Multiplication	Parallel Computing
	Blocked Algorithms for Memory-Efficient Decompositions	Memory & Numerical Optimization
	Low-Rank Approximation for Matrix Compression	Memory & Numerical Optimization
	Mixed-Precision Numerical Linear Algebra Calculation	Memory & Numerical Optimization

Real-World Applications

Distributed ML:
- Multi-GPU Multi-node matrix operations
- Distributed training of neural networks using model parallelism
- Fully Sharded Data Parallel (FSDP) for memory-efficient distributed parameter updates
Computer vision:
- Transformation vision data matrices
NLP:
- Word embedding via matrix factorization (SVD)
Recommender systems:
- Collaborative filtering with matrix factorization
Robotics:
- Kinematic chain transformations using basis changes
Graphics:
- Projection matrices in 3D rendering pipelines

Linear Algebra

Core Knowledge

Linear Algebra Fundamentals

Numerical Stability

Matrix Factorization Techniques

Implementation Mastery

Key Questions

Common Pitfalls

Off-by-one errors in matrix indexing operations

Floating-point precision errors

Mishandling matrix dimensions

O(n³) complexity matrix operations when O(n²) solutions exist

Numerical instability in decomposition algorithms

Inefficient handling of sparse matrix storage formats

Extended Questions

Real-World Applications

Core Knowledge

Linear Algebra Fundamentals

• Common Matrix/vector operations

• Matrix/vector operation complexities and edge cases

• Special matrix properties (diagonal, orthogonal, sparse)

• Rank-nullity theorem applications

• Geometric interpretation of linear transformations

• Pseudoinverse implementations for rank-deficient systems

Numerical Stability

• Floating-point error propagation analysis

• Condition number sensitivity

• Regularization techniques for ill-posed problems

Matrix Factorization Techniques

• Decomposition tradeoffs (SVD vs QR vs Cholesky)

• Iterative eigenvalue algorithms

Implementation Mastery

• Batched operation optimizations

• Parallel computation patterns

• Multi-threaded matrix multiplication algorithm

• Sparse storage formats (CSR/CSC performance characteristics)

• Memory layout implications (row/column-major)

• Cache-aware matrix algorithms

Key Questions

Common Pitfalls

Off-by-one errors in matrix indexing operations

Floating-point precision errors

Mishandling matrix dimensions

O(n³) complexity matrix operations when O(n²) solutions exist

Numerical instability in decomposition algorithms

Inefficient handling of sparse matrix storage formats

Extended Questions

Real-World Applications