#L06 · Linear Algebra & Manifolds · Lv.2View in 3D Mind-Map

Gaussian Elimination & Linear Systems (Ax = b)

Nine Chapters, Gauss & Gabriel Cramer · 1750 CE

A\mathbf{x} = \mathbf{b} \implies x_i = \frac{\det(A_i)}{\det(A)}

[A \mid \mathbf{b}] \xrightarrow{\text{RREF}} [I \mid \mathbf{x}]

Solves systems of simultaneous linear equations Ax = b via systematic row reduction (Gaussian elimination) and determinant ratios (Cramer's rule).

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Coeff a₁

Const c₁

Coeff a₂

Const c₂

\begin{cases}2x + y = 5\\ 1x - y = 1\end{cases} \implies (x, y) = (2.00,\; 1.00)

연립방정식 교점 해 (x, y)

(2.00, 1.00)

계수 행렬식 det(A)

-3.00

행렬 랭크 rank(A)

2 (유일해)

Prerequisite Path (14 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#G01Pythagorean Theorem

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

STEP 4#G03Pi, Radian Measure & Circular Sector

l = r\theta

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#G022D/3D Euclidean Distance Formula

d(P

STEP 8#G05Law of Sines & Law of Cosines

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

STEP 9#G04Unit Circle Trigonometry & Identities

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

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#L033D Cross Product & Surface Normal Vector

\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y

STEP 12#L04Matrix Multiplication & Linear Transformation

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

STEP 13#L05Determinant & Matrix Inverse

\det(A) = ad - bc

STEP 14#L06Gaussian Elimination & Linear Systems (Ax = b)

A\mathbf{x} = \mathbf{b} \implies x_i = \frac{\det(A_i)}{\det(A)}

Geometric & Intuitive Insight

2차원 평면에서 두 일차방정식은 두 개의 직선이며, 연립방정식의 해 (x, y)는 두 직선이 교차하는 단 하나의 교점입니다. 두 직선이 평행하면(det(A)=0) 교점이 없거나 무수히 많습니다.

Step-by-Step Derivation

Step 1.1단계: 연립방정식 a₁x + b₁y = c₁, a₂x + b₂y = c₂ 를 행렬 형태 Ax = b 로 표기합니다.
Step 2.2단계: 첫째 식에 b₂를 곱하고 둘째 식에 b₁을 곱해 빼면 y가 소거되어 (a₁b₂ - b₁a₂)x = c₁b₂ - b₁c₂
Step 3.3단계: det(A) = a₁b₂ - b₁a₂ ≠ 0 일 때 x = det(A_x)/det(A), y = det(A_y)/det(A) 로 해가 구해집니다.

AI & Coding Real-World Use

SPICE circuit simulation, structural mechanics solvers, and Least Squares Normal Equations

#Linear Solver Ax=b#SPICE Circuit Sim#Normal Equations

const det = a1*b2 - b1*a2; const x = (c1*b2 - b1*c2)/det; const y = (a1*c2 - c1*a2)/det;

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