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The following is from an exercise in Gilbert Strang's Linear Algebra and its Applications:

Suppose $A$ has eigenvalues $0,3,5$ with independent eigenvectors $u,v,w$.
Find a particular solution to $Ax = v+w$. Find all solutions.

It is not difficult to find that the particular solution can be $\frac{1}{3}v+\frac{1}{5}w$. Here is my question:

How should I find all solutions for the equation?

If the equation is $Ax = 0$, one needs to find a basis for the null space of $A$. However in this case, the right hand side is $v+w$.

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    Hint 2: what do you think is a basis for the null space of $A$?2011-07-27

1 Answers 1

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Thanks to Geoff Robinson's hint, one should first find a basis for the null space. In this case, $\{u\}$ can be the basis. Then any solution can be written as $x=ku+\frac{1}{3}v+\frac{1}{5}w\qquad k\in {\mathbb R}.$

This may be the standard way to solve the non-homogeneous linear equation.

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    Re: "This may be the standard way". In fact, this **is** the standard way. 1. Solve the homogeneous problem. 2. Find one solution $x_p$ of the non-homogeneous problem. 3. Every solution is of the form $x_h + x_p$ where $x_h$ is a solution of the homogeneous problem.2011-07-28