homogenous equations vs nonhomogenous equations

See also: Linear Algebra, column space, null space, solution space, variation of parameters

The Comparison: $AX=0$ vs. $AX=b$

Here is the structured reconstruction of the concepts on the board:

Feature	Homogeneous System	Nonhomogeneous System
Form	$AX=0$	$AX=b$
Consistency	Always consistent. (Why? Because the zero vector $\vec{0}$ is always a solution. This is called the “trivial solution”.)	Maybe consistent or inconsistent. (If $b$ is outside the column space of $A$, there is no solution.)
Solution Set	Is a Subspace. The set of solutions forms the Null Space, $N(A)$. It passes through the origin.	Is NOT a Subspace (if $b\ne 0$). It does not contain the zero vector, and adding two solutions yields a vector that is no longer a solution.
General Solution	$X_h=c_1{v}_1+\cdots +c_k{v}_k$ (Linear combination of basis vectors)	$X_g=X_p+X_h$ (Particular Solution + Homogeneous Solution)

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The “Deep Dive”: Why isn’t $AX=b$ a Subspace?

The board presents a mini-proof in the third row that is very important for understanding “Linearity.”

For a set of vectors to be a Subspace, it must satisfy two main rules:

1. Closure under Addition: If $x_1$ and $x_2$ are solutions, then $x_1+x_2$ must be a solution.

2. Closure under Scalar Multiplication: If $x$ is a solution, then $c\cdot x$ must be a solution.

Why Homogeneous ($AX=0$) works:

If $A{x}_1=0$ and $A{x}_2=0$, then:

$A(x_1+x_2)=A{x}_1+A{x}_2=0+0=0$

The sum is still a solution.

Why Nonhomogeneous ($AX=b$) fails:

The board explicitly writes this derivation. If $A{x}_1=b$ and $A{x}_2=b$, look what happens when you add them:

$A(x_1+x_2)=A{x}_1+A{x}_2=b+b=2b$

Since $2b\ne b$ (assuming $b\ne 0$), the sum is not a solution. The set is “broken” under addition.

Geometric Intuition:

• The solution to $AX=0$ is a line (or plane) passing through the origin.

• The solution to $AX=b$ is that same line (or plane), but shifted away from the origin by the vector $X_p$. Since it doesn’t go through the origin, it can’t be a subspace. (Mathematically, we call this an Affine Space).

The “Engineering” Connection: $X_g=X_p+X_h$

The final row ($X_g=X_p+X_h$) is a concept you will see repeatedly, not just in Linear Algebra, but in Differential Equations and Control Theory.

The logic is:

To describe all possible solutions to the complex problem ($AX=b$), you only need:

1. One single specific working example ($X_p$, the Particular solution).

2. All the ways you can have “zero effect” ($X_h$, the Null space).

You essentially say: “Here is one path to the target $b$, and from there, I can move anywhere along the ‘zero’ directions without messing up my target.”

Bkz: Matematik • Engineering • Feynman