Building Nonhomogeneous Solutions From One Particular Solution

From One Solution to More Solutions

A nonhomogeneous system \(\mathbf{A}\mathbf{x}=\mathbf{b}\) asks for vectors whose matrix product is a specified vector \(\mathbf{b}\). If we already know one solution, it can serve as a starting point for constructing others. The key is to change that solution only by a vector that the matrix sends to zero: such a change does not alter the product.

Throughout, let \(\mathbf{A}\in\mathbb{R}^{m\times n}\) and \(\mathbf{b}\in\mathbb{R}^m\). A solution \(\mathbf{x}_p\in\mathbb{R}^n\) of \(\mathbf{A}\mathbf{x}=\mathbf{b}\) is called a particular solution. A vector \(\mathbf{z}\in\mathbb{R}^n\) is a homogeneous solution when \(\mathbf{A}\mathbf{z}=\mathbf{0}\); equivalently, \(\mathbf{z}\in\operatorname{Nul}\mathbf{A}\). The subscripts distinguish the chosen particular solution from other possible solutions.

Proof. By linearity of matrix-vector multiplication, the matrix applied to the constructed vector is

Thus the constructed vector satisfies the required equation. \(\square\)

The theorem gives a reliable way to produce solutions: start with one known solution and add any linear combination of known homogeneous solutions. It makes a one-way guarantee: every vector built this way works. Whether this construction accounts for every solution is a separate question, so we will not assume that conclusion here. In computations, the homogeneous solutions are often found by describing the null space with free parameters, as in the earlier material on homogeneous systems.

One Homogeneous Direction

This example illustrates how free parameters from the homogeneous equation become adjustable directions from the particular solution. A choice such as \(s=1,t=0\) gives \(\begin{bmatrix}1\\1\\0\end{bmatrix}\); choosing \(s=0,t=1\) gives \(\begin{bmatrix}4\\0\\1\end{bmatrix}\). Both are different vectors, and the direct multiplication check confirms that both produce the same right-hand side.

Combining and Checking Solutions

It is not necessary to use a single homogeneous direction at a time. The theorem allows any real coefficients on any finite list of homogeneous vectors. The coefficients can be zero, negative, or positive; the only condition is that each direction being added must be sent to \(\mathbf{0}\) by the same matrix \(\mathbf{A}\).

What the Construction Does—and Does Not—Require

The operation depends on two checks, both of which matter. First, \(\mathbf{x}_p\) must solve the nonhomogeneous equation with the specified right-hand side. Second, every added direction must solve the homogeneous equation for the same coefficient matrix. If either check fails, linearity does not produce the desired conclusion. The argument also works when a homogeneous direction is multiplied by any real scalar, because the matrix sends every scalar multiple of a zero-product vector to zero.

This construction is useful when a system has already yielded one solution and its homogeneous solutions have been described using free parameters. Rather than solving the nonhomogeneous equations again for each new answer, we can keep the particular solution fixed and adjust its homogeneous components. Each adjustment leaves the image under \(\mathbf{A}\) unchanged, so all constructed vectors have the same right-hand side.

It is important not to confuse a valid construction with a completeness claim. The theorem proved here guarantees that adding homogeneous solutions produces solutions. It does not, by itself, establish that every solution of the nonhomogeneous system must arise in this way. A proof of that stronger statement needs an additional argument. For the present method, verify the two inputs and use the theorem to certify each output.

Common Mistake: A Sign Error in a Direction

In the first worked example, the homogeneous equation is \(z_1+2z_2-z_3=0\). If we choose \(z_2=1\) and \(z_3=0\), this equation requires \(z_1=-2\), not \(z_1=2\). A mistaken sign leads to the proposed direction \(\widetilde{\mathbf{z}}=\begin{bmatrix}2\\1\\0\end{bmatrix}\), but

Therefore \(\widetilde{\mathbf{z}}\) is not a homogeneous solution and cannot be used as a direction in the construction. For instance, adding it to \(\mathbf{x}_p=\begin{bmatrix}3\\0\\0\end{bmatrix}\) gives \(\begin{bmatrix}5\\1\\0\end{bmatrix}\), whose product is \(\begin{bmatrix}7\\14\end{bmatrix}\), not \(\mathbf{b}=\begin{bmatrix}3\\6\end{bmatrix}\). The correction is to check each proposed direction by computing \(\mathbf{A}\mathbf{z}\) before adding it to a particular solution.

A second practical check is to keep vector dimensions visible. If \(\mathbf{A}\) has \(n\) columns, then both the particular solution and every homogeneous direction must be in \(\mathbb{R}^n\); their sum is then defined and remains in the domain of \(\mathbf{A}\). The right-hand side, by contrast, must be in \(\mathbb{R}^m\), matching the number of rows of \(\mathbf{A}\).