Table of Contents

Linear Algebra in AI/ML Coding

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 like recommendation systems and computer vision.

Core Knowledge: Linear Algebra in AI/ML

  • Linear Algebra Fundamentals

  • Numerical Stability

  • Matrix Factorization Techniques

  • Implementation Mastery and Optimisation

Key Coding Interview Questions

StatusQuestionCategory
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

Advanced Linear Algebra Coding Questions

StatusQuestionCategory
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

Frequently Asked Follow-up Questions

Discuss and compare the pros and cons of different sparse matrix storage formats (CSR/CSC) for different use cases.
CSR (Compressed Sparse Row) is row-oriented: perfect for fast row slices and SpMV operations on CPU
  • Pros: contiguous row data, efficient row-wise dot products, smaller index array when rows dominate.
  • Cons: column access is indirect; transpose requires re-packing.

CSC (Compressed Sparse Column) flips the orientation: ideal for column operations and many linear-algebra kernels on GPU
  • Pros: accelerated column slices, efficient for solving Ax = b with sparse direct solvers.
  • Cons: row access is indirect; less friendly for SpMV if the right-hand side changes frequently.

Guideline: choose CSR for iterative methods and row-major algorithms, CSC for factorisation-heavy solvers or if you repeatedly access columns.
Explain how to implement a batched matrix multiplication algorithm with optimal memory usage.
1. Tiling: load an M×K tile of A and a K×N tile of B into shared/L2 cache; multiply; accumulate into a tile of C.
2. Loop ordering: iterate over K in the outer loop to reuse tiles of A and B across the batch.
3. Stride-1 memory access: store batches in [batch, row, col] (row-major) or use array-of-pointers to avoid padding waste.
4. Vectorisation: issue SIMD/FMA instructions inside the innermost kernel.
5. Frameworks: on GPU, use cuBLAS BatchedGEMM; on CPU, call MKL’s cblas_sgemm_batch for free scheduling and pre-fetching.
The result is near-optimal arithmetic intensity and minimal memory traffic per floating-point operation.
Discuss the trade-offs between different matrix factorization techniques (SVD, QR, Cholesky) for different use cases.
• SVD: most stable; reveals rank and singular values; O(mn²). Use for dimensionality reduction, pseudoinverse, noisy data.
• QR: cheaper O(mn²/2); orthogonal Q enables least-squares; numerically robust via Householder. Pick for real-time regression.
• Cholesky: fastest O(n³/6), but requires positive-definite symmetric matrices. Ideal for covariance inversion and Kalman filters.
Rule of thumb: SVD when you need accuracy and rank info, QR for over-determined systems, Cholesky for SPD solves where speed matters most.
Compare different multi-threaded matrix multiplication algorithms and their performance implications.
• Naïve row/column partitioning: trivial to implement; suffers cache thrashing and load imbalance.
• Blocked (cache-aware) GEMM + OpenMP: splits matrices into tiles; excellent locality; scales on shared-cache CPUs.
• NUMA-aware partitioning: binds tiles to memory nodes; avoids remote access latency on multi-socket servers.
• Strassen-like algorithms: reduce arithmetic work (O(n^{2.81})) but introduce irregular memory access; rarely worth the overhead below ~1024×1024.
• SUMMA/Scalable GEMM: used in distributed memory; overlaps communication and computation.
Interview tip: emphasise blocked + OpenMP for practicality, and mention NUMA pinning for servers with >2 sockets.
Discuss floating-point error propagation in matrix operations and how to mitigate it.
Error model: each arithmetic op has relative error ≤ ε (machine epsilon). For a matrix algorithm, the forward error grows with the condition number κ(A).

Mitigations
• Scale or precondition A to lower κ.
• Prefer Householder QR to Gram–Schmidt; it halves error amplification.
• Use pivoting in LU/Cholesky when data are near-singular.
• Accumulate inner products with Kahan summation or pairwise reduction.
• Switch to mixed-precision: compute in FP16/FP32, refine with one FP64 Newton step.
Summarise by relating error ≤ κ · ε and showing how each technique attacks κ or ε directly.