ex1.ai

Example 83: Basis + coordinates: one-hot lookup and PCA change-of-basis

Example 84: Linear maps: forward/backward as matrix products

Example 85: Similarity geometry: dot vs cosine in embedding retrieval

Example 86: Orthogonality in regression: residual â column space

Example 87: Rank/null space: many parameters, same predictor

Example 88: Eigenvectors in ML: power iteration for dominant PCA direction

Example 89: PSD in ML: covariance and kernel Gram matrices

Example 8: Power iteration for dominant eigenvector (PCA direction)

Example 90: SVD and conditioning: why normal equations are risky

Example 91: PCA bookkeeping: EVR and reconstruction error

Example 92: Least squares: lstsq vs normal equations

Example 93: Solving systems: Cholesky factor reuse

Example 94: Conditioning: small perturbations â big solution changes

Example 95: Sparse matrices: adjacency COO + O(nnz) matvec

Example 96: Transformers: attention as QK^T then AV

Example 97: Vector spaces + subspace projection (embeddings â zero-mean)

Example 98: Span in ML: Xw and attention as weighted sums

Example 99: Basis + coordinates: one-hot lookup and PCA change-of-basis

Example 9: PSD checks: covariance and kernel Gram matrices

Vector Spaces & Subspaces