Solution Sets of Linear Systems

A solution set is simply the collection of every answer a system has, and when there are infinitely many, that collection has a shape you can picture: a line, a plane, or something bigger. All we're really saying in this lesson is: find one answer, then add on every way of "changing nothing" (every solution of $A\vec x = \vec 0$), and you've got all the answers. That's good because it describes infinitely many solutions with just a couple of vectors. The formula $\vec x = \vec p + \vec v_h$ has just two pieces: $\vec p$ (one particular solution, any one you like, so $A\vec p = \vec b$) and $\vec v_h$ (any solution of $A\vec x = \vec 0$, so adding it doesn't change what $A$ produces). It looks abstract, but it's one answer plus some wiggle room; bear with it.

Homogeneous systems. A system is called homogeneous when its right side is all zeros, $A\vec x = \vec 0$. These always have the trivial solution $\vec x = \vec 0$, since $A\vec 0 = \vec 0$ no matter what $A$ is. They have nontrivial solutions exactly when there's a free variable. That comes straight from row reduction: a free variable can be set to anything, so you can set it to something nonzero.

Parametric vector form. Write the solution as a combination of vectors with the free variables as weights: $\vec x = s\vec u + t\vec v$ describes a plane through the origin. Notice this is just a span from earlier in the unit: the solutions of $A\vec x = \vec 0$ are the span of $\vec u$ and $\vec v$.

Nonhomogeneous systems. The solution set of $A\vec x = \vec b$ (if consistent) is $\vec p + \{\text{solutions of } A\vec x = \vec 0\}$: a parallel copy, shifted off the origin. Think about why that works. If $A\vec p = \vec b$ and $A\vec v_h = \vec 0$, then by linearity $A(\vec p + \vec v_h) = \vec b + \vec 0 = \vec b$, so $\vec p + \vec v_h$ is a solution too. It works in reverse as well: if $\vec x$ is any solution, then $A(\vec x - \vec p) = \vec b - \vec b = \vec 0$, so $\vec x - \vec p$ is one of the homogeneous solutions. Every solution is $\vec p$ plus something $A$ sends to $\vec 0$, nothing more.

If you've taken differential equations, this should look familiar. It's the same structure as $y = y_p + y_c$ for differential equations: one particular solution plus everything that solves the version with zero on the right.

A common mistake. It's tempting to think the solutions of $A\vec x = \vec b$ form a span, like the homogeneous ones, because they're written with the same kind of vectors. They don't: a span always contains $\vec 0$, and $\vec 0$ doesn't solve $A\vec x = \vec b$ when $\vec b \ne \vec 0$. The solution set is a shifted copy that misses the origin. And it doesn't matter which particular solution you use for $\vec p$; any one of them gives the same shifted set.

There's an animation further down where you slide along a solution line like this one.

In circuit analysis and structural engineering, the homogeneous solutions are the "free modes" of a system, and the particular solution is the response to the applied loads.