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



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


because it is linear



how to apply it to vector and produce co vector?please help me

dato datuashvili
  • 8,684
  • 15
  • 76
  • 137

1 Answers1


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 $g(V)=3-5+6-8=-4$.

About a couple of your sub-questions:

I want to calculate its corresponding covector

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.)

how to apply it to vector and produce a covector?

I think you mean how to apply a covector to a vector and produce a scalar.

Chris Culter
  • 26,935
  • 4
  • 46
  • 102