EigenVectors system of differentials

[dp182]

Homework Statement

Find the general solution for the following systems of equations
( 2 0 )
( 0 2 )

Homework Equations

(A-Ix)
c1e^at[]+c2e^bt[]

The Attempt at a Solution

when attempting to find the eigenvalues I come up with (2) so plugging back into get the vectors you come up with a zero matrix how do i get vectors from it do i just pick any ones

 

[HallsofIvy]

For any vector, $\begin{bmatrix}x \\ y\end{bmatrix}$, in R²,
$$\begin{bmatrix}2 & 0 \\ 0 & 2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}2x \\ 2y\end{bmatrix}$$

so that any vector is an eigenvector.

That's why you can just choose any two independent vectors as a basis for the eigenspace.

Of course
$$\begin{bmatrix}1 \\ 0 \end{bmatrix}$$
and
$$\begin{bmatrix}0 \\ 1 \end{bmatrix}$$
are the "standard" choice.