Let $e=\{e_1,e_2,…,e_n\} $ be a basis for the complex vector space X. Is there a general way of finding a basis for X when regarded as a real vector space?
We know that for every x in X there are complex $z_i$ such that: $$ x= \sum_{j=1}^n z_i e_i =(z_1,…,z_n)_e $$
and $z_i$ not all 0. Intuitively I'd say that when regarded as a real vector space X has dimension $2n$ , so we should use all the $e_i$ plus another n vectors which we'll find from the expression of x, but I can't seem to find how to even start doing it. Perhaps I'm wrong, do correct me 🙂 Thanks in advance.
Edit: Are the vectors of e even linearly independent in the real vector space? Are any linearly independent vectors of the complex vector space, independent in the real one?
Best Answer
A simple-to-find basis is $$ e_1, i\cdot e_1, e_2, i\cdot e_2,\ldots, i\cdot e_n $$ And vectors in a complex vector space that are complexly linearly independent, which means that there is no complex linear combination of them that makes $0$, are automatically real-linearly dependent as well, because any real linear combination is a complex linear combination, and we've established that those do not exist.
In fact, we may do the same as above, and take our set of vectors, duplicate it, multiply each element in the duplicate by $i$, and we will have a twice as large, real-linearly independent set.