Why linear systems appear in engineering
Many engineering models reduce to several equations that must be satisfied at the same time. These equations
may come from force balance, current balance, flow balance, compatibility, or stiffness laws.
\[
A\mathbf{x}=\mathbf{b}
\]
Here \(A\) is the coefficient matrix, \(\mathbf{x}\) is the vector of unknowns, and \(\mathbf{b}\)
is the right-side vector.
Matrix form of a system
A system such as
\[
\begin{aligned}
a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=b_1\\
a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n&=b_2\\
&\vdots\\
a_{m1}x_1+a_{m2}x_2+\cdots+a_{mn}x_n&=b_m
\end{aligned}
\]
can be written compactly as
\[
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix}
\begin{bmatrix}
x_1\\
x_2\\
\vdots\\
x_n
\end{bmatrix}
=
\begin{bmatrix}
b_1\\
b_2\\
\vdots\\
b_m
\end{bmatrix}.
\]
Augmented matrix
Gaussian elimination is usually performed on the augmented matrix:
\[
[A\mid\mathbf b]
\]
This places the coefficient matrix and the right-side vector in one table, so row operations can be applied
to the whole system at once.
Gaussian elimination
Gaussian elimination uses row operations to simplify the system. The allowed operations are:
- Swap two rows.
- Multiply a row by a nonzero constant.
- Add a multiple of one row to another row.
These operations do not change the solution set. They only rewrite the system in a simpler form.
RREF and solution reading
In reduced row echelon form, pivot columns identify leading unknowns. If every unknown has a pivot and the
system is consistent, the solution is unique.
\[
\operatorname{rref}([A\mid\mathbf b])
\]
A row such as
\[
\begin{bmatrix}
0 & 1 & 0 & \mid & 5
\end{bmatrix}
\]
means \(x_2=5\).
Rank test
The rank test tells whether a system has one solution, no solution, or infinitely many solutions.
\[
\operatorname{rank}(A)
=
\operatorname{rank}([A\mid\mathbf b])
=
n
\]
If this holds, the system has a unique solution.
\[
\operatorname{rank}(A)
<
\operatorname{rank}([A\mid\mathbf b])
\]
means the system is inconsistent and has no solution.
\[
\operatorname{rank}(A)
=
\operatorname{rank}([A\mid\mathbf b])
<
n
\]
means the system is dependent and has infinitely many solutions.
Residual check
After a numerical solution is found, the residual checks how well it satisfies the original equations:
\[
\mathbf r
=
A\mathbf{x}-\mathbf b.
\]
A small residual means the solution fits the original system well. A large residual may indicate rounding,
poor conditioning, or incorrect input.
Statics and force equilibrium
In statics, equations often come from component balance:
\[
\sum F_x=0,
\qquad
\sum F_y=0,
\qquad
\sum F_z=0.
\]
If unknown forces act along known directions, their direction components become coefficients in the matrix.
The solved unknowns give force magnitudes.
Truss member forces
At a truss joint, equilibrium equations can be written for the member forces. A positive solution usually
means the assumed direction is correct. A negative solution usually means the actual force acts opposite
the assumed direction.
Circuit analysis
Circuit nodal analysis also produces linear systems. Using Kirchhoff's current law, each node equation
can be written in terms of node voltages:
\[
G\mathbf{v}=\mathbf{i}.
\]
Here \(G\) is often a conductance matrix, \(\mathbf{v}\) is the vector of unknown node voltages, and
\(\mathbf{i}\) is the current-source vector.
Structural stiffness systems
Structural stiffness equations often take the form
\[
K\mathbf{u}=\mathbf{F}.
\]
Here \(K\) is the stiffness matrix, \(\mathbf{u}\) is the displacement vector, and \(\mathbf{F}\) is
the load vector.
Common mistakes
- Putting coefficients in the wrong columns.
- Changing signs when moving terms to the right side.
- Forgetting that a negative force solution means opposite assumed direction.
- Using inconsistent units across rows.
- Assuming every system has a unique solution.
- Ignoring rank diagnostics when equations are dependent.
- Not checking the residual after solving.
