I am currently using Lay's Lineair algebra and its functions, on page 316.
On this page, I have the following problem. One page earlier is stated that a multiplication x' = Ax (where A is a matrix and x is a vector) has the following two solutions (not the only two):
$$ x_1(t) = \mathbf{v}e^{\lambda t}, x_2(t) = \overline{\mathbf{v}}e^{\overline{\lambda}t} (6) $$ Where v is the eigenvector of A. Soo the two solutions given here are $x_1(t)$ and $\overline{x_1}(t)$, which I get. Now the book says the following: "Fortunately, the real and imaginary parts of $x_1(t)$ are (real) solutions, because they are lineair combinations of the solutions in (6)". And then it states two solutions: $$ y_1 = 1/2 [x_1 + \overline{x_1}], y_2 = 1/(2i)[x_1 - \overline{x_1}] $$
So the book claims (and is correct) that these two are solutions to Ax = x'. But I cant seem to prove it. How would I prove this?
$T(x)=Ax$ is a linear transformation so if $u,v$ are solutions to $Ax=b$, then so is $1/2(u+v)$, as $T(1/2(u+v))=1/2(Au+Av)=1/2(b+b)=b$