MA25C01-LINEAR ALGEBRA🚀 LINEAR COMBINATION & SYSTEM OF LINEAR EQUATIONS | ENGINEERING MATHEMATICS

Linear Algebra · Study Notes

Linear Combinations &
Systems of Linear Equations

Homogeneous Systems Non-Homogeneous Systems Rouché–Capelli Theorem

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Systems of linear equations are the backbone of linear algebra. Whether you are solving engineering models, analysing data, or studying abstract vector spaces, understanding when solutions exist and how many there are is fundamental. This post walks through both homogeneous and non-homogeneous systems with full definitions, key theorems, and a geometric perspective.

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Part I The Homogeneous System

What is a Homogeneous System?

A system of linear equations is homogeneous if every constant term is zero. In general form, a homogeneous system of \(m\) equations in \(n\) unknowns looks like:

\[ \begin{aligned} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n &= 0 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n &= 0 \\ \vdots \hspace{2.5cm} & \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n &= 0 \end{aligned} \]

Matrix Representation

The system compresses elegantly into the matrix equation:

\[A\mathbf{x} = \mathbf{0}\] where \(A = [a_{ij}]_{m \times n}\) is the coefficient matrix, \(\mathbf{x}\) is the column vector of unknowns, and \(\mathbf{0}\) is the zero vector.

Trivial vs Non-Trivial Solutions

• Trivial Solution: The zero vector \(\mathbf{x} = \mathbf{0}\) — i.e., \(x_1 = x_2 = \cdots = x_n = 0\) — is always a solution to any homogeneous system \(A\mathbf{x} = \mathbf{0}\).
• Non-Trivial Solution: Any solution where at least one unknown is non-zero.

"Every homogeneous system is consistent by default — the interesting question is whether it has something beyond the trivial."

Key Theorems

\(AX = 0\) has only the trivial solution (\(X = 0\)) if and only if \(\lvert A \rvert \neq 0\).

\(AX = 0\) has a non-trivial solution (\(X \neq 0\)) if and only if \(\lvert A \rvert = 0\).

Existence of Non-Trivial Solutions. A homogeneous system \(A\mathbf{x} = \mathbf{0}\) of \(m\) equations in \(n\) unknowns has a non-trivial solution if and only if: \[ \operatorname{rank}(A) < n \iff \text{non-trivial solutions exist.} \]

Remark. If \(m < n\) (fewer equations than unknowns), the system always has a non-trivial solution.

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Part II The Non-Homogeneous System

Definition

A system is non-homogeneous when at least one constant term is non-zero:

\[ \begin{aligned} a_{11}x_1 + \cdots + a_{1n}x_n &= b_1 \\ a_{21}x_1 + \cdots + a_{2n}x_n &= b_2 \\ \vdots \hspace{2cm} &\ \vdots \\ a_{m1}x_1 + \cdots + a_{mn}x_n &= b_m \end{aligned} \]

where \(b_1, b_2, \ldots, b_m\) are real numbers, not all zero.

Matrix Representation & Augmented Matrix

The compact form is \(A\mathbf{x} = \mathbf{b}\). The augmented matrix appends the constants:

\[ [A \mid \mathbf{b}] = \left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \end{array}\right] \]

Rouché–Capelli Theorem

\(A\mathbf{x} = \mathbf{b}\) is consistent (has at least one solution) if and only if: \[ \operatorname{rank}(A) = \operatorname{rank}([A \mid \mathbf{b}]) \]

Types of Solutions

Let \(r = \operatorname{rank}(A) = \operatorname{rank}([A \mid \mathbf{b}])\) and \(n\) be the number of unknowns.

1. If \(r = n\) → unique solution.
2. If \(r < n\) → infinitely many solutions (with \(n - r\) free variables).
3. If \(\operatorname{rank}(A) < \operatorname{rank}([A \mid \mathbf{b}])\) → no solution (inconsistent).

Quick Reference

Condition	Outcome	Status
\(\operatorname{rank}(A) \neq \operatorname{rank}([A|\mathbf{b}])\)	No solution	Inconsistent
\(\operatorname{rank}(A) = \operatorname{rank}([A|\mathbf{b}]) = n\)	Unique solution	Consistent
\(\operatorname{rank}(A) = \operatorname{rank}([A|\mathbf{b}]) < n\)	Infinitely many solutions	Consistent

Geometric Interpretation

For \(m\) equations in \(n\) unknowns, each equation defines a hyperplane in \(\mathbb{R}^n\). The solution set is precisely the intersection of these hyperplanes.

• Unique solution: All hyperplanes meet at a single point.
• Infinitely many solutions: The hyperplanes share a line, plane, or higher-dimensional affine subspace.
• No solution: No common intersection exists — the hyperplanes are parallel or skew.

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