Vector with complex values

Question

This may seem a very basic question in linear algebra, but I hope somebody will clear my doubt. I am aware of the vector with real values and am able to perform various operations over it. But I fail to understand what a vector with complex values will signify. For example, if I have a vector with 3 real values, I take it as the vector in R^3 space. What about if some of the values of this vector are complex? What would those complex values signify? Will it still be a vector in three-dimensional space? Furthermore, how can I project this vector over another vector?

Answer 1

The main thing is to understand the concept of a vector space. If you take a look at the definition it says, that it is regarded as a vector space over a field. The entries of the vectors (vectors as elements of the specific vector space you are considering) are in this specific field.

So if you consider a vector with 3 real entries you can indeed regard this as a vector in $\mathbb{R}^3$ over the field of the real numbers $\mathbb{R}$.

Now since the reals are contained in the complex numbers, you may also regard this vector over the vector field of the complex numbers if you like. Considering the question to what happens if some entries are complex. In this case you consider your field to be the complex numbers $\mathbb{C}$, and you may choose your vector space to be $\mathbb{C}^3$ if you consider a vector with 3 entries (here it does not depend if some or all of them are complex valued since we work over $\mathbb{C}$).

As Osama was pointing out in the comments, you should try to understand the isomorphism $$\mathbb{C} \cong \mathbb{R}^2$$ or more general $$\mathbb{C}^n \cong \mathbb{R}^{2n}.$$

So if you want to consider the vector with 3 entries and some of the entries may be complex, you consider is as an element in $\mathbb{C}^3$ and if you regard this vector space over $\mathbb{C}$ you would have a vector space of dimension 3. From the above isomorphism if you regard it over $\mathbb{R}$ it would be a vector space of dimension 6.

Regarding the last question, you would proceed in the standard way, see for instance https://en.wikipedia.org/wiki/Vector_projection.