Applications of Linear Systems
Circuits, chemistry, traffic, data — every balance law in engineering is a row in a matrix. Gaussian elimination is the universal solver.
From World to Matrix
- Identify the unknown quantities
- Write down every balance or conservation law
- Each law becomes one row of A
- Each unknown becomes one column
- Solve Ax = b with Gaussian elimination
Electrical Circuits
KCL: currents entering a node = currents leaving.
KVL: voltage drops around any loop sum to zero.
Balancing Chemistry
Conservation of each element type gives one linear equation in the stoichiometric coefficients.
Homogeneous system Ax = 0. One free variable (scaling). The null space gives the balanced coefficients: 2, 7, 4, 6.
Network Flow
- Inflow = outflow at every node
- One equation per junction
- One unknown per edge (street / pipe / link)
- Total inflow = total outflow is redundant
- Free variables = routing flexibility
Least Squares Fitting
More data points than unknowns → overdetermined Ax = b, no exact solution. Minimize ||Ax − b||² instead.
Linear Regression
Fitting y = mx + c to n data points: build A with rows [xᵢ 1], b with entries yᵢ. The normal equations give the best-fit slope and intercept.
Signal Filters
FIR filter: y[n] = h₀x[n] + h₁x[n−1] + … + h_{L−1}x[n−L+1]
More specs than coefficients? Use least squares for the best-fit filter.
Key Takeaways
- Conservation laws → rows of a matrix
- KCL + KVL gives currents from circuit topology
- Chemical balancing lives in the null space
- Network flow: consistent, often underdetermined
- Overdetermined? Use normal equations AᵀAx̂ = Aᵀb
- FIR filter design = linear system in coefficients
Module 4: Eigenvalues
You've mastered setting up and solving linear systems. Next: the special vectors that a matrix simply stretches — eigenvectors — and the deep structure they reveal.