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I've got what should be a simple question. I have a function f(x,y) = 3x + 2y

The question asks to write this function as a sum of dual vectors. Any help on where to begin?

Philip Bruen
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    Can you add to your question what you think a dual vector is? Perhaps you can find a basis of dual vectors and express $f$ as a linear combination of the elements of that basis? – gt6989b Aug 14 '19 at 04:15
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    This is one of those cases where it's probably important to quote the question verbatim. There may be a requirement in the question that is more important than you think. – Chris Culter Aug 14 '19 at 04:19
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    "where to begin" is with the definition of dual vector (or covector as it says in the title). What do those terms mean to you? – Gerry Myerson Aug 14 '19 at 04:21
  • a) Suppose we have some linear function f : R 2 → R defined as f(x, y) = 3x + 2y. Write f as a sum of dual vectors. – Philip Bruen Aug 14 '19 at 04:22
  • I think a dual vector is a complimentary vector that satisfies v^i*(vj) = kroenecker delta (i,j) – Philip Bruen Aug 14 '19 at 04:26
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    What's a complimentary vector? (or even a complementary vector?) What do $v^i$ and $v_j$ stand for? In any event, if you don't know what a dual vector is, then it's no wonder you can't solve the problem. So, the place for you to begin is by looking through whatever study materials you have to find out exactly what's meant by "dual vector". – Gerry Myerson Aug 14 '19 at 13:28
  • Are you still here, Philip? – Gerry Myerson Aug 16 '19 at 00:13
  • Yes I'm here a dual vector takes input of a vector and outputs a real number. It pulls out the component of the vector that aligns with it. It also gives Kronecker delta being 1 when components of both components of both the dual vector and the vector is the same and 0 otherwise – Philip Bruen Aug 17 '19 at 03:35
  • The word "component" only makes sense when you have specified a basis. Since no basis is specified, I take it we are assuming by default the "standard basis" $e_1=(1,0)$, $e_2=(0,1)$. Then the dual vectors are, as nole writes in an answer, $e^1$ and $e^2$ given by $e^1(x,y)=x$ and $e^2(x,y)=y$. So all you're being asked to do is to write $f(x,y)=3x+2y$ as a sum (though I would rather say, linear combination) of $e^1$ and $e^2$. So, can you do that? – Gerry Myerson Aug 19 '19 at 12:46
  • Gone away again, Philip? – Gerry Myerson Aug 20 '19 at 13:03
  • I'm voting to close this question as off-topic because OP has abandoned it. – Gerry Myerson Aug 22 '19 at 13:00

1 Answers1

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Try the dual vectors of the canonical basis $\{e_1,e_2\}$ of $\mathbb{R}^2$. Observe that $e^1(x,y)=x$ and $e^2(x,y)=y$.

nole
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