# understanding dual vectors and dual spaces

please help me with number example to understand better what is a co-vector and what is a dual vector space,as i know it is related to fact,that it eats vector,which means that it takes vector and calculates scalar,also as i know there should be used linear function,or functions which satisfies following relations

1.$f(v_1+v_2)=f(v_1)+f(v_2)$

2,$f(a*v)=a*f(v)$

so let take some vector ,let say

$V=v_1,v_2,v_3,v_4...v_n$ ok let take numbers $V=(3,4,5,6,7,8)$ and i want to calculate it's corresponding co vector,how could i solve this problem,let take this function

$f(x)=5*x$

because it is linear

$f(x_1+x_2)=f(x_1)+f(x_2)$

2.$f(a*x)=a*f(x)$

• i read all this definition,but could not understand,that why i have posted to make explanation easiest – dato datuashvili Sep 1 '13 at 18:33
• could you give me example with simply example? – dato datuashvili Sep 1 '13 at 18:33
• i am going to bad,and please just i need example without formal notations,it makes impossible for me to understand topic,so please with real numbers – dato datuashvili Sep 1 '13 at 18:38
• Dear dato, Have you seen this answer? Regards, – Matt E Sep 1 '13 at 20:22

The function $f(x)=5x$ is not a covector for the space of vectors of the form $V=v_1,v_2,v_3,v_4,\ldots,v_n$. (I'm assuming that $n>1$.) A covector must input a vector and output a scalar. Your function $f$ inputs a vector and outputs another vector.

Here's a better example. Let $g(v_1,v_2,v_3,v_4,v_5,v_6)=v_1-v_3+v_4-v_6$. Then $g$ is a covector on $\mathbb R^6$, and given your vector $V=(3,4,5,6,7,8)$, we have $g(V)=3-5+6-8=-4$.

Given a vector $V$, there is generally no covector corresponding to $V$. (There will be such a correspondence if you have an inner product, but don't worry about that until you get to it.)