An exercise for eigenvalues and eigenvectors

Question

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 $v+w$.

Hint: any two solutions differ by an element of the null space — Geoff Robinson, Jul 27 '11 at 23:17
Hint 2: what do you think is a basis for the null space of $A$? — Srivatsan, Jul 27 '11 at 23:57

score 6 · Accepted Answer · answered Jul 27 '11 at 23:55

6

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.

answered Jul 27 '11 at 23:55

6

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. – t.b. Jul 28 '11 at 00:53

An exercise for eigenvalues and eigenvectors

1 Answers1

Linked