Linear Algebra: Linear Models in Business, Science, and Engineering

Linear Equations and Electrical Networks

• Current flow in simple electrical networks can be modeled using systems of linear equations.

• System Components:

  • Voltage Source: Such as a battery, which forces a current of electrons to flow through a network.

  • Resistor: A component that "uses up" some of the voltage as current passes through it.

• Ohm’s Law: The "voltage drop" across a resistor is defined by the formula:     $V = R \times I$

  • $V$: Voltage, measured in volts.

  • $R$: Resistance, measured in ohms ($\Omega$).

  • $I$: Current flow, measured in amperes (commonly referred to as amps).

Kirchhoff’s Voltage Law (KVL)

• Definition: The algebraic sum of $R \times I$ voltage drops in one direction around a closed loop is equal to the algebraic sum of the voltage sources in that same direction around the loop.

• Application (Example 2): Determining loop currents in a specific network (e.g., Figure 1):

  • For Loop 1, current flows through three resistors. The total sum of the $R \times I$ voltage drops is calculated as the sum of those resistances multiplied by the loop current.

  • Interaction with other loops: In branch AB, current from Loop 2 also flows. If the direction for branch AB in Loop 1 is opposite to the flow chosen for Loop 2, the voltage drop from Loop 2 is subtracted ($-R_2 \times I_2$).

  • Equation Construction: The algebraic sum of all $R \times I$ drops for a loop is set equal to the loop's voltage source. For example, if Loop 1 has a $+30$ volt source, the resulting equation is modeled based on Kirchhoff's Voltage Law.

  • The Loop 2 Equation: Includes a term representing the flow of Loop 1 current through branch AB (negative because the flow is opposite to Loop 2's direction), a term representing the sum of all resistances in Loop 2 multiplied by its current, and a term for Loop 3 current flowing through branch DC in the opposite direction.

  • The Loop 3 Equation: Follows the same principles of summing resistances and accounting for shared branches with negative terms for opposing current flows.

Solving Network Systems and Matrix Models

• Solution Process: Loop currents are determined by solving a system of linear equations (referred to as system (3) in the text).

• Augmented Matrix: Performing row operations on the augmented matrix derived from the system leads to the specific values for the current (in amperes) for each loop.

• Vector Equation Representation:

  • The system can be viewed as a vector equation: $I_1 \mathbf{r}_1 + I_2 \mathbf{r}_2 + I_3 \mathbf{r}_3 = \mathbf{v}$.

  • The first entry of each vector concerns the first loop, the second entry concerns the second loop, and so on.

  • Resistance Vector Entries: The vector for each current (e.g., $\mathbf{r}_1$ for current $I_1$) lists the resistances in various loops through which that specific current flows. A resistance value is written as a negative number when the current flows against the flow direction of another loop.

• Matrix Form (Matrix Ohm’s Law):

  • The network is described by the equation: $R \mathbf{i} = \mathbf{v}$.

  • $R$: The resistance matrix.

  • $\mathbf{i}$: The current vector.

  • $\mathbf{v}$: The voltage vector.

  • Note on Matrix R: If all loop currents are chosen to flow in the same direction (e.g., counter-clockwise), all entries off the main diagonal of matrix $R$ will be negative.

Linearity and Superposition in Electrical Systems

• The linearity of the electrical model is evident in the matrix equation $R \mathbf{i} = \mathbf{v}$. If the voltage vector is doubled ($2\mathbf{v}$), the current vector must also double ($2\mathbf{i}$).

• Superposition Principle: The solution of the general equation is the sum of the solutions of the specific equations: $R\mathbf{i} = \mathbf{v}_1$, $R\mathbf{i} = \mathbf{v}_2$, …, $R\mathbf{i} = \mathbf{v}_n$.

  • Each specific equation corresponds to the circuit containing only one voltage source, with all other sources replaced by wires to close the loops.

• Foundational Linear Laws: The model is linear because both Ohm’s Law (voltage drop proportional to current) and Kirchhoff's Law (sum of drops equals sum of sources) are linear.

• Branch Current Determination:

  • If only one loop current passes through a branch, the branch current equals that loop current.

  • If multiple loop currents pass through a branch, the branch current is the algebraic sum of those loop currents (Kirchhoff’s Current Law).

Difference Equations and Dynamic Systems

• In fields like ecology, economics, and engineering, dynamic systems that change over time are modeled using measurements at discrete time intervals.

• State Vectors: A sequence of vectors $\mathbf{x}_0, \mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_k, \dots$ represents the system's status. The entries in $\mathbf{x}_k$ provide data about the system's state during the $k$-th measurement.

• Linear Difference Equation (Recurrence Relation): This occurs if a matrix $A$ exists such that:     $\mathbf{x}_1 = A \mathbf{x}_0$     $\mathbf{x}_{k+1} = A \mathbf{x}_k$

• Computability: Provided the initial state $\mathbf{x}_0$ is known, one can iteratively compute $\mathbf{x}_1$, then $\mathbf{x}_2$, and so on.

Population Migration Model

• Conceptual Model: Demographers track the movement of groups (e.g., movement between a city and its suburbs) over a period of years.

• Initial Conditions:

  • Let the year 2020 be the initial time ($k = 0$).

  • $c_0$: Population of the city in 2020.

  • $s_0$: Population of the suburbs in 2020.

  • $\mathbf{x}_0 = \begin{pmatrix} c_0 \\ s_0 \end{pmatrix}$: Initial population vector.

• Migration Rates:

  • From City: Each year, $5\%$ of the city's population moves to the suburbs, while $95\%$ remains in the city.

  • From Suburbs: Each year, $3\%$ of the suburban population moves to the city, while $97\%$ remains in the suburbs.

• Mathematical Distribution after 1 Year:

  • City population in 2021: $c_1 = 0.95 c_0 + 0.03 s_0$

  • Suburban population in 2021: $s_1 = 0.05 c_0 + 0.97 s_0$

• Migration Matrix (M): The movement can be expressed in matrix form:     $\mathbf{x}_1 = M \mathbf{x}_0$

  • Where matrix $M$ is defined by the table of migration percentages:         $M = \begin{pmatrix} 0.95 & 0.03 \\ 0.05 & 0.97 \end{pmatrix}$

    • Columns represent "From": City (Col 1), Suburbs (Col 2).

    • Rows represent "To": City (Row 1), Suburbs (Row 2).

• General Migration Formula: If migration percentages remain constant over time, the state follows the general linear difference equation:     $\mathbf{x}_{k+1} = M \mathbf{x}_k$

  • This sequence of vectors describes the total regional population distribution over a period of years.