1. ## Linear Algebra notation

Hello, I am having a hard time understanding what is meant by this notation:

2. Are the following vector spaces?

(iii) $V= \lbrace f: \Re \longrightarrow C : f(-t) = \overline{f(t)}, \forall t \in \Re \rbrace$, (over $C$ and with the usual addition and scalar multiplication of functions).

(The $C$ is the complex numbers; I'm not sure how to do that in Latex)

The part that's comfusing me is the overline... does that mean that $\overline{f(t)}$ is a vector? Do the reals map to the complexes as a vector because it's an ordered pair $(a,b)$ with $a+bi$ as a complex number? Thank you for any help!

2. Originally Posted by sfitz
Hello, I am having a hard time understanding what is meant by this notation:

2. Are the following vector spaces?

(iii) $V= \lbrace f: \Re \longrightarrow C : f(-t) = \overline{f(t)}, \forall t \in \Re \rbrace$, (over $C$ and with the usual addition and scalar multiplication of functions).

(The $C$ is the complex numbers; I'm not sure how to do that in Latex)

The part that's comfusing me is the overline... does that mean that $\overline{f(t)}$ is a vector? Do the reals map to the complexes as a vector because it's an ordered pair $(a,b)$ with $a+bi$ as a complex number? Thank you for any help!
$V=\left\{f: \mathbb{R}\mapsto\mathbb{C} \left.\right| f\!\left(-t\right)=\overline{f\!\left(t\right)},~\forall~t\i n\mathbb{R}\right\}$

The first bit, as you've seen, refers to a mapping from the Real to the Complex plane. The last condition is not referring to a vector per se (Yes, complex numbers can be seen as a vector), but the overbar (in $\mathbb{C}$) refers to the complex conjugate.

Let's say for example, we have $f\!\left(t\right)=a+ti,~a,t\in\mathbb{R}$

Thus, $f\!\left(-t\right)=a+(-t)i=a-ti=\overline{f\!\left(t\right)}$

I leave it up to you to see if it is a vector space or not.

I hope this helps clarify things wrt notation.