Author Jonathan David
Author Jonathan David
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Introductory Example: Linear Models in Economics and Engineering

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Linear Models in Economics and Engineering

Linear models are foundational in economics and engineering, and their analysis often requires linear algebra techniques, such as solving systems of equations, matrix representation, and vector operations.

Economics Example: Supply and Demand

In economics, the relationship between supply and demand can be modeled linearly. Consider the system:

    \[P = a - bQ_d \quad \text{(Demand equation)}\]

    \[P = c + dQ_s \quad \text{(Supply equation)}\]

Rewriting these equations as a linear system with Q_d = Q_s = Q:

    \[\begin{aligned} P - a + bQ &= 0, \\ P - c - dQ &= 0. \end{aligned}\]

This system can be expressed in matrix form:

    \[\begin{bmatrix} 1 & b \\ 1 & -d \end{bmatrix} \begin{bmatrix} P \\ Q \end{bmatrix} = \begin{bmatrix} a \\ c \end{bmatrix}.\]

To solve for the equilibrium price (P) and quantity (Q), we can use techniques like Gaussian elimination or matrix inversion, provided the determinant of the coefficient matrix is nonzero:

    \[\det\begin{bmatrix} 1 & b \\ 1 & -d \end{bmatrix} = -b - d \neq 0.\]

Engineering Example: Force and Displacement

In engineering, Hooke’s Law relates force and displacement linearly:

    \[F = kx.\]

For a system of n springs in parallel, the total force and displacement relationship can be written as:

    \[\mathbf{F} = \mathbf{K}\mathbf{x},\]

where:

  • \mathbf{F} = \begin{bmatrix} F_1 \\ F_2 \\ \vdots \\ F_n \end{bmatrix} is the force vector,
  • \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} is the displacement vector,
  • \mathbf{K} = \text{diag}(k_1, k_2, \dots, k_n) is the diagonal matrix of spring constants.

If the system is in equilibrium under external forces, we solve for \mathbf{x} using:

    \[\mathbf{x} = \mathbf{K}^{-1}\mathbf{F},\]

assuming \mathbf{K} is invertible.

Linear Algebra Techniques in Practice

Both examples highlight key linear algebra techniques:

  • Matrix representation of linear systems,
  • Solving linear systems using Gaussian elimination, determinants, or matrix inverses,
  • Understanding matrix properties, such as invertibility and determinants, to assess system solvability.

These methods enable the efficient analysis of complex systems in economics, engineering, and beyond.

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