---
id: wiki-2026-0508-linear-algebra
title: Linear Algebra
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Linear Algebra for ML, Vectors and Matrices, SVD, Eigendecomposition]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [math, linear-algebra, numpy, svd, eigen, pca, ml-foundations]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack: { language: Python, framework: numpy/torch }
---

# Linear Algebra

## 매 한 줄
> **"매 ML은 행렬 곱셈이다"**. Vector·Matrix·Tensor 위에서 projection, rotation, decomposition으로 모든 게 표현된다.

## 매 핵심
### 매 객체
- Scalar, Vector ∈ ℝⁿ, Matrix ∈ ℝᵐˣⁿ, Tensor (high-rank).
- Norms: ‖x‖₁ (sparsity), ‖x‖₂ (energy), ‖x‖∞.
- Inner product, outer product, cosine similarity.

### 매 연산 핵심
- Matmul C=AB. shape (m,k)·(k,n)=(m,n).
- Transpose, inverse, pseudoinverse (Moore-Penrose).
- Determinant (scaling), trace (diagonal sum, eigen sum).
- Rank: 독립 column 수. low-rank → 압축 가능.

### 매 분해
- **Eigendecomposition** A = QΛQ⁻¹ (square, diagonalizable). PCA covariance.
- **SVD** A = UΣVᵀ (any matrix). 가장 일반적.
- **QR** Gram-Schmidt. least squares 안정.
- **Cholesky** A = LLᵀ (symm. PD). 빠른 solve, GP, Kalman.
- **LU** general solve.

### 매 ML 응용
1. **PCA**: covariance eigen / data SVD → top-k.
2. **Linear regression**: x̂ = (XᵀX)⁻¹Xᵀy 또는 SVD pseudoinverse.
3. **Recommendation MF**: A ≈ UVᵀ.
4. **Word embeddings**: LSA SVD, word2vec implicit MF.
5. **Attention**: softmax(QKᵀ/√d)V — 전부 matmul.

## 💻 패턴
### NumPy 핵심
```python
import numpy as np
A = np.random.randn(4, 3); x = np.random.randn(3)
y  = A @ x                            # matmul
G  = A.T @ A                          # 3x3 Gram
inv = np.linalg.inv(G)
sol = np.linalg.solve(G, A.T @ y)     # 안정적인 normal eq
```

### SVD & truncated rank-k
```python
U, S, Vt = np.linalg.svd(A, full_matrices=False)
k = 2
A_k = U[:, :k] @ np.diag(S[:k]) @ Vt[:k]
```

### Eigen / PCA
```python
X = np.random.randn(1000, 10)
Xc = X - X.mean(0)
cov = Xc.T @ Xc / (len(X) - 1)
vals, vecs = np.linalg.eigh(cov)              # symmetric → eigh
order = np.argsort(-vals)
PCs = vecs[:, order[:3]]
Z = Xc @ PCs                                  # (N, 3)
```

### Least squares 4가지
```python
# 1) normal eq (불안정)
b1 = np.linalg.inv(A.T @ A) @ A.T @ y
# 2) solve (better)
b2 = np.linalg.solve(A.T @ A, A.T @ y)
# 3) lstsq (SVD-based, 가장 안정)
b3, *_ = np.linalg.lstsq(A, y, rcond=None)
# 4) pseudoinverse
b4 = np.linalg.pinv(A) @ y
```

### einsum (general tensor)
```python
# batch matmul (B,M,K)·(B,K,N) → (B,M,N)
C = np.einsum("bmk,bkn->bmn", X, Y)
# attention scores
scores = np.einsum("bqd,bkd->bqk", Q, K) / np.sqrt(Q.shape[-1])
```

### Norms / cosine
```python
def cosine(a, b):
    return (a @ b) / (np.linalg.norm(a) * np.linalg.norm(b) + 1e-12)
```

### PyTorch (autograd)
```python
import torch
A = torch.randn(4, 3, requires_grad=True)
loss = (A @ x - y).pow(2).sum()
loss.backward()                       # dL/dA
U, S, Vt = torch.linalg.svd(A, full_matrices=False)
```

### Cholesky (GP / Kalman)
```python
L = np.linalg.cholesky(K + 1e-6 * np.eye(n))   # K SPD + jitter
alpha = np.linalg.solve(L.T, np.linalg.solve(L, y))
```

## 매 결정 기준
| 작업 | 함수 |
|---|---|
| 일반 solve | `np.linalg.solve` |
| Least squares | `np.linalg.lstsq` (SVD) |
| Symm. eigen | `eigh` |
| 일반 eigen | `eig` |
| 일반 분해 | `svd` |
| SPD solve 빠르게 | Cholesky |
| Sparse 큰 행렬 | `scipy.sparse.linalg` (eigsh, svds) |
| GPU | torch.linalg / cupy |

**기본값**: 정확도/안정성은 SVD/Cholesky, 속도는 solve, 빠른 prototype은 lstsq.

## 🔗 Graph
- 부모: [[Mathematics]], [[Numerical-Methods]]
- 변형: [[SVD]], [[Eigendecomposition]], [[QR-Decomposition]], [[Cholesky]]
- 응용: [[PCA]], [[Linear-Regression]], [[Latent-Semantic-Analysis-LSA]], [[Attention]]
- Adjacent: [[Tensor]], [[Numpy]], [[Optimization]], [[Calculus]]

## 🤖 LLM 활용
**언제**: 식 유도, einsum 변환, 함수 선택, shape debug.
**언제 X**: numerical conditioning / iterative solver tuning은 전문가.

## ❌ 안티패턴
- `inv(A) @ b` 대신 `solve(A, b)` 안 씀
- Symm 행렬에 일반 `eig` (느림+정확도)
- Large dense에 raw SVD (메모리) → randomized/truncated
- Loop matmul (vectorize)
- 차원/축 mismatch — `einsum`으로 명시
- Float32 누적 오차 (PCA covariance) → float64 또는 standardize

## 🧪 검증 / 중복
- Verified (Strang, Trefethen NLA, NumPy/PyTorch docs). 신뢰도 A.
- 중복: 없음.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — einsum, lstsq, Cholesky 패턴 추가 |