# Math Help - Easy Linear Algebra Question

1. ## Easy Linear Algebra Question

Hey guys! New year new math class. I'm in Linear Algebra and its been quite a few years since ive taken the intro to linear alg course so i'm a little rusty and don't have that book any more. In the following question i'm not quite certain how to approach this problem. The problem states:

A real-valued function f defined on the real line is called an even functions if f(-t)=f(t) for each real number t. Prove that the set of even functions defined on the real line with the operations of addition and scalar multiplication defined in the previous example is a vector space.

Previous Example: Let S be any nonempty set and F be and field, ad let F(S,F) denote the set of all functions from S to F. Two functions f and g in F(S,F ) are called equal if f(s)=g(s) for each s in S. The set F(S,F) is a vector space with the operations of addition and scalar multiplication defined for f,g in F(S,F) and c in F by: (f+g)(s)=f(s)+g(s) and (cf)(s)=c[f(s)] for each s in S.

I'm just not sure what to do :/. Thanks in advance!

2. This will work:
$(f+\alpha g)(-x)=f(-x)+ \alpha g(-x)= f(x)+ \alpha g(x)=(f+\alpha g)(x)$

3. Using what you wrote you could show that if f(-x)= f(x) and g(-x)= g(x) then (f+ g)(-x)= f(-x)+ g(-x)= f(x)+ g(x)= (f+ g)(x) and then that $(\alpha f)(-x)= \alpha f(-x)= \alpha f(x)= (\alpha f)(x)$. What Plato did was combine those two into one equation.

(I am using (f+ g) and $(\alpha f)$ to mean the functions formed by adding f and g and by multiplying f by the number $\alpha$. Of course, those are defined by $(f+ g)(x)= f(x)+ g(x)$ and $(\alpha f)(x)= \alpha f(x)$.)