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I'm having trouble with this:

Let's say we have a homogeneous system of linear equations (HSLE), where all the coefficients are real numbers. Let $f_1, f_2, \dots , f_m$ be the fundamental system of solutions (FSS) for all the real solutions.

Question: How do you prove that $f_1, f_2, \dots , f_m$ is also the fundamental system of solutions for all the complex solutions.

I know that the solutions of the HSLE should form a vector space whose basis is the (FSS), but in this case the FSS is the basis for only the real solutions vector space. I somehow need to show that this FSS is the basis for both real and complex solutions vector space, but how?

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    Ok, well give us an example equation and we'll show you how it works2012-04-18

1 Answers 1

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The space of solutions is an $m$-dimensional $\mathbb C$-linear subspace. We know that it contains the $m$-dimensional $\mathbb R$-linear subspace spanned by $f_j$. The rest follows from a lemma:

Lemma. There is a unique $m$-dimensional $\mathbb C$-linear subspace $M_{\mathbb C}\subset \mathbb C^n$ containing a given $m$-dimensional $\mathbb R$-linear subspace $M\subset \mathbb R^n$. It is given by $M_{\mathbb C}=\{u+iv: u,v\in M\}$ and is called the complexification of $M$.

Proof is straightforward: by its definition, $M_{\mathbb C}$ must be contained in any $\mathbb C$-linear subspace that contains $M$. Since its dimension is $m$, the claim follows.

When $M=\langle f_j\rangle_{\mathbb R}$, $M_{\mathbb C}$ obviously agrees with $\langle f_j\rangle_{\mathbb C}$.