Linear Equations

46. Linear Equations

Linear Equation

𝑦 = 𝑎_{1} 𝑥_{1} + 𝑎_{1} 𝑥_{2} + \dots + 𝑎_{𝑛} 𝑥_{𝑛}

Whether a system of linear equations has at least one solution

Example

Consistent System

\begin{matrix} 𝑥 + 𝑦 = 3 \\ 𝑥 - 𝑦 = 1 \end{matrix}

This system has a unique solution

(𝑥, 𝑦) = (2, 1)

Inconsistent System

\begin{matrix} 𝑥 + 𝑦 = 3 \\ 𝑥 + 𝑦 = 5 \end{matrix}

This system is inconsistent (equations contradict each other, no solution can satisfy both)

Whether the equations in the system provide unique and non-redundant information about the variables

Example

Independent Equations

\begin{matrix} 𝑥 + 𝑦 = 3 \\ 𝑥 - 𝑦 = 1 \end{matrix}

Neither equation can be derived from the other (they provide unique information and intersect at a single point)

Dependent Equations

\begin{matrix} 𝑥 + 𝑦 = 3 \\ 2 𝑥 + 2 𝑦 = 6 \end{matrix}

Second equation is just a multiple of the first equation (they describe the same line)

Example

\begin{matrix} 3 𝑥 + 2 𝑦 = 6 (Equation 1) \\ 6 𝑥 + 4 𝑦 = 12 (Equation 2) \end{matrix}

Unique Solution (Consistent and Independent):

No Solution (Inconsistent):

Infinitely Many Solutions (Consistent and Dependent):

System of Equations

\begin{matrix} 𝑎_{11} 𝑥_{1} + 𝑎_{12} 𝑥_{2} + \dots + 𝑎_{1 𝑛} 𝑥_{𝑛} = 𝑏_{1} \\ 𝑎_{21} 𝑥_{1} + 𝑎_{22} 𝑥_{2} + \dots + 𝑎_{2 𝑛} 𝑥_{𝑛} = 𝑏_{2} \\ ⋮ \\ 𝑎_{𝑚 1} 𝑥_{1} + 𝑎_{𝑚 2} 𝑥_{2} + \dots + 𝑎_{𝑚 𝑛} 𝑥_{𝑛} = 𝑏_{𝑚} \end{matrix}

Matrix Representation

Coefficient vector ( $Α$ )

𝐴 = [\begin{matrix} 𝑎_{11} & 𝑎_{12} & \dots & 𝑎_{1 𝑛} \\ 𝑎_{21} & 𝑎_{22} & \dots & 𝑎_{2 𝑛} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{2 𝑚 1} & 𝑎_{𝑚 2} & \dots & 𝑎_{𝑚 𝑛} \end{matrix}]

Variable vector ( $𝑥$ )

𝑥 = [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ ⋮ \\ 𝑥_{𝑛} \end{matrix}]

Constant vector ( $𝑏$ )

𝑏 = [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \\ ⋮ \\ 𝑏_{𝑚} \end{matrix}]

Matrix equation

𝐴 𝑥 = 𝑏

linear_eq.py

from scipy.linalg import solve

X = np.array([
  [1, 1, 1],
  [2, -1, 3],
  [3, 4, -1]
])

Y = np.array([6, 14, 1])

intersection_point = solve(X, Y)