Specifically, I'm trying to solve the following problem:
Let $V$ be a finite-dimensional complex vector space and $e_1,
\cdots,e_n\in V$ be a basis. Show the following:(i) addition $V\times V\rightarrow V$ and restriction of scaling $\mathbb{C}\times V$ to $\mathbb{R}\times V$ make $V$ a real vector space;
(ii) the vectors $e_1,\cdots,e_n,ie_1,\cdots,ie_n$ are a basis for $V$ as a real vector space.
Generally speaking, I intuitively understand this as the decomplexification of a vector space where we "ignore" multiplication by complex values. I'm struggling to formally solve this. Here's my attempt:
(i): The restriction of scaling implies that $V_{\mathbb{C}}\ni u+iv\mapsto u+v$. The desired result follows.
(ii): Suppose $\lambda_1e_1+\cdots+\lambda_ne_n+\lambda_1ie_1+\cdots+\lambda_nie_n=0$ for some $\lambda_1,\cdots,\lambda_n\in\mathbb{R}$. Then $\lambda_1e_1+\cdots+\lambda_ne_n=-i(\lambda_1e_1+\cdots+\lambda_ne_n)$, which implies that $\lambda_1=\cdots=\lambda_n=0$. Hence, $e_1,\cdots,e_n,ie_1,\cdots,ie_n$ are linearly independent in $V_{\mathbb{R}}$.
The questions I have are as follows:
- What does the addition $V\times V\rightarrow V$ contribute to part (i) aside from the fact that for $V$ to be a real vector space, addition must be defined?
- What is a better way to express what the restriction of scaling is doing? I used the \mapsto symbol, but there isn't really a mapping going on.
- Aside from questions 1 and 2, am I missing anything to finish solving part (i)?
- Is my justification that $e_1,\cdots,e_n,ie_1,\cdots,ie_n$ are linearly independent valid? It feels sketchy to repeat scalars.
- How can I show that $e_1,\cdots,e_n,ie_1,\cdots,ie_n$ span $V_{\mathbb{R}}$? It seems to follow directly from the fact that $e_1,\cdots,e_n$ are a basis for $V_{\mathbb{C}}$ and the "decomposition" of complex vectors into real parts by restriction of scaling.
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