System of linear equation

Classification of Systems

A system of linear equations can be:

1. Non-singular (Unique solution): Each equation provides new information. Geometrically:
   • 2D → two lines intersect at one point.
   • 3D → three planes intersect at one point.
2. Singular (No unique solution):
   • Redundant (Infinite solutions): Equations overlap same line/plane repeated.
   • Contradictory (No solution): Equations conflict parallel lines/planes.

Key fact: Constants (right-hand side values) shift the solution, but singularity depends only on the coefficient matrix.

Method 1 Manual Elimination

We can manipulate equations in three ways without changing the solution set:

1. Multiply an equation by a non-zero constant.
2. Swap two equations.
3. Add/subtract a multiple of one equation to another.

These are exactly the row operations we’ll use in matrix form.

Solving by elimination

$$\{\begin{array}{l}2x+y=5(1)\\ x-y=1(2)\end{array}$$

Step 1: Eliminate one variable

From (2): $x-y=1$ Multiply (2) by 2: $2x-2y=2$ Subtract from (1):

$$(2x+y)-(2x-2y)=5-2$$ $$3y=3$$ $$y=1$$

Step 2: Back-substitution

From (2): $x-1=1\Rightarrow x=2$ Solution: $(x,y)=(2,1)$ — unique, so non-singular.

Gaussian Elimination (Matrix Form)

$$\{\begin{array}{l}2x+y=5\\ x-y=1\end{array}$$

Coefficient matrix:

$$\left[\begin{array}{cc}2& 1\\ 1& -1\end{array}\right]$$

Augmented matrix (add constants as last column):

$$\left[\begin{array}{ccc}2& 1& 5\\ 1& -1& 1\end{array}\right]$$

Row operations

1. Row swap: $R_i\leftrightarrow R_j$
2. Row scale: $R_i\to k{R}_i$, $k\ne 0$
3. Row replacement: $R_i\to R_i+k{R}_j$

All preserve singularity and the solution set.

2.5.2 Goal forms

Row Echelon Form (REF):

• All non-zero rows above zero rows.
• Pivot (first non-zero number) in each row is to the right of the pivot above.
• Zeros below pivots.

Reduced Row Echelon Form (RREF):

• REF + each pivot = 1.
• Zeros above and below pivots.

RREF directly shows the solution.

Example in matrix form

$$\left[\begin{array}{ccc}2& 1& 5\\ 1& -1& 1\end{array}\right]$$

Step 1: Swap rows so pivot is 1 at top:

$R_1\leftrightarrow R_2$:

$$\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& 5\end{array}\right]$$

Step 2: Eliminate below pivot:

$R_2\to R_2-2R_1$:

$$\left[\begin{array}{ccc}1& -1& 1\\ 0& 3& 3\end{array}\right]$$

Step 3: Scale pivot in $R_2$:

$R_2\to \frac{1}{3}{R}_2$:

$$\left[\begin{array}{ccc}1& -1& 1\\ 0& 1& 1\end{array}\right]$$

Step 4: Eliminate above pivot in col 2:

$R_1\to R_1+R_2$:

$$\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 1\end{array}\right]$$

This is RREF. Solution is $x=2,y=1$.

Detecting Singular Systems in Gaussian Elimination

1. Redundant (Infinite solutions):

Last row becomes $00|0$ — trivial equality $0=0$.

2. Contradictory (No solution):

Last row becomes $00|c$ with $c\ne 0$ — false statement.

Example — Redundant:

$$\{\begin{array}{l}x+y=2\\ 2x+2y=4\end{array}$$

Row reduction → last row: $00|0$.

Example — Contradictory:

$$\{\begin{array}{l}x+y=2\\ x+y=3\end{array}$$

Row reduction → last row: $00|1$.

Rank of a Matrix

Definition: The rank is the number of pivots in the REF form.

• Full rank: Rank = number of rows → non-singular.
• Less than full rank: Singular.

Example:

$$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

Rank = 2 → non-singular.

2.8 Determinants and Singularity

For 2×2:

$$\det \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=ad-bc$$

• det ≠ 0 → non-singular.
• det = 0 → singular.

For 3×3, formula is more complex but same principle holds.

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