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Math Help - Linear Algebra notation

  1. #1
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    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!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by sfitz View Post
    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.
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