#L04 · Linear Algebra & Manifolds · Lv.2View in 3D Mind-Map

Matrix Multiplication & Linear Transformation

Arthur Cayley · 1858 CE

(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}

AB \neq BA \quad (\text{Non-commutative})

Computes each entry (i,j) as the dot product of row i of A and column j of B, geometrically representing the composition of two linear transformations.

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A₁₁

A₁₂

B₁₁

B₂₁

\begin{bmatrix}2 & 1\\ -1 & 2\end{bmatrix}\begin{bmatrix}1\\3\end{bmatrix} = \begin{bmatrix}5.0\\5.0\end{bmatrix}

합성 성분 (AB)₁₁ = 1행·1열

5.000

합성 성분 (AB)₂₁ = 2행·1열

5.000

변환 스케일 배율

2.236

Prerequisite Path (11 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 2#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 3#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 4#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 5#L01Vector Operations, L1/L2 Norms & Normalization

\|\mathbf{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^2}

STEP 6#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 7#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 8#G022D/3D Euclidean Distance Formula

d(P

STEP 9#G05Law of Sines & Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

STEP 10#L02Dot Product & Cosine Similarity

\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

STEP 11#L04Matrix Multiplication & Linear Transformation

(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}

🚀 Next Unlocks (Derived Formulas) (6)

#L05Determinant & Matrix Inverse #L06Gaussian Elimination & Linear Systems (Ax = b)#L072D/3D Rotation Matrix & Orthogonal Group SO(3)#L08Eigenvalues & Eigenvectors (Av = λv)#L09Singular Value Decomposition (SVD) & PCA #P10Softmax Function & Stochastic Gradient Descent

Geometric & Intuitive Insight

왜 행렬 곱셈은 같은 자리끼리 곱하지 않고 '행 × 열'로 복잡하게 곱할까요? 벡터 x에 변환 B를 먼저 가하고(Bx) 이어서 변환 A를 가한 A(Bx) 의 최종 결과 식을 정리해보면 정확히 (AB)x 의 행×열 내적 규칙이 나오기 때문입니다!

Step-by-Step Derivation

Step 1.1단계: 벡터 x를 y = Bx 로 변환하면 y_k = ∑_j B_{kj} x_j 입니다.
Step 2.2단계: 다시 y를 z = Ay 로 변환하면 z_i = ∑_k A_{ik} y_k = ∑_j (∑_k A_{ik} B_{kj}) x_j 가 됩니다.
Step 3.3단계: 따라서 합성 변환 행렬 C = AB 의 (i, j) 성분은 C_{ij} = ∑_k A_{ik} B_{kj} 로 유일하게 결정됩니다.

AI & Coding Real-World Use

90% of NVIDIA GPU Tensor Core compute in Deep Learning (Y = XW) and 3D graphics MVP pipelines

#GPU Tensor Cores GEMM#Deep Learning Y=XW#3D MVP Pipeline

const matMul2 = (A:number[][], B:number[][]) => [[A[0][0]*B[0][0]+A[0][1]*B[1][0], A[0][0]*B[0][1]+A[0][1]*B[1][1]]];

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