Linear Algebra Concepts for AI using CUDA ToolKit | Download

A comprehensive reference for core linear algebra concepts used in Artificial Intelligence and Machine Learning.

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1. Scalars, Vectors, Matrices, and Tensors

• Scalar: A single number (e.g., learning rate, bias).
• Vector: Ordered 1D array (e.g., feature vector of an image).
• Matrix: 2D array (e.g., dataset with rows = samples, columns = features).
• Tensor: Generalization to higher dimensions (e.g., RGB image → 3D tensor).

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2. Vector Operations

• Addition & Subtraction
• Scalar Multiplication
• Dot Product (Inner Product) → measures similarity (used in cosine similarity for NLP):
  $$\mathbf{a} \cdot \mathbf{b} = \sum_i a_i b_i$$
• Cross Product (only in 3D) → used in geometry and physics:
  $$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}$$

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3. Matrix Operations

• Addition, Subtraction, Scalar Multiplication
• Matrix Multiplication → core operation in neural networks:
  $$C = A \cdot B$$
• Transpose →
  $$A^T$$
• Identity Matrix ($I$) → like multiplying by 1
• Inverse Matrix ($A^{-1}$) → solving linear systems
• Determinant → volume, singularity check:
  $$\det(A)$$
• Normalization → scaling rows/columns to unit norm, often applied before feeding data to models

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4. Norms & Distance Measures

• Euclidean Distance (L2):
  $$|x - y|_2 = \sqrt{\sum_i (x_i - y_i)^2}$$
• Manhattan Distance (L1):
  $$|x - y|_1 = \sum_i |x_i - y_i|$$
• Cosine Similarity:
  $$ \cos(\theta)=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|;|\mathbf{b}|} $$

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5. Systems of Linear Equations

• Represented as:
  $$Ax = b$$
• Solutions: unique, infinite, or none
• Solved with matrix inverses or decomposition

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6. Linear Transformations

• Matrix as a transformation (e.g., rotation, scaling, projection)
• Crucial for understanding how data moves through neural networks

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7. Eigenvalues and Eigenvectors

• $$Av = \lambda v$$
• Eigenvectors = directions of variance
• Eigenvalues = magnitude of variance along the eigenvector
• Used in PCA (Principal Component Analysis), spectral clustering, etc.

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8. Matrix Decompositions

• LU Decomposition → efficient solving of linear systems
• QR Decomposition → orthogonality
• Singular Value Decomposition (SVD) → foundation of PCA, used in recommendation systems (e.g., Netflix, YouTube)

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9. Orthogonality & Projections

• Orthogonal vectors → dot product = 0:
  $$\mathbf{u} \cdot \mathbf{v} = 0$$
• Orthonormal basis → like standard axes in 3D (x, y, z)
• Projection → mapping data onto lower dimensions:
  $$\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}$$
• Key in dimensionality reduction

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10. Vector Spaces & Subspaces

• Span, Basis, Rank, Null space
• Rank = number of independent features:
  $$\text{rank}(A)$$
• Null space = solutions to:
  $$Ax = 0$$

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11. Special Matrices

• Diagonal → simple scaling
• Symmetric → real eigenvalues
• Orthogonal → preserves length & angle
• Sparse → many zeros, used in NLP

References

• Book: Discrete Mathematics and Its Applications (7th Edition)
• Video: A Deep Dive into the Latest HPC Software
• Video: Deep Dive into Math Libraries
• Video: Scale Python and NumPy Performance with Legate