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I'm just stuck on this question. How can I represent the additive inverse of all continuous functions?

The additive inverse: For every $\overrightarrow{u}$ in V, there is a vector V denoted by $\overrightarrow{-u}$ such that $\overrightarrow{u}$ + ($\overrightarrow{-u}$) = $\overrightarrow{0}$.

Any help is appreciated.

This is a solution I found to a similiar problem earlier:

Describe the additive inverse of the vector space $P_3$ where $P_3$ is the set of all polynomials of degree 3 or below. Solution: $-(a_0 + a_1x + a_2x^2 + a_3x^3) = -a_0 - a_1x - a_2x^2 - a_3x^3$

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Inverse or identity? And what do you mean by describe? Describe how, in what language or context? Remember that complete strangers on the internet don't have your homework sheet, and class notes to work with... –  Asaf Karagila Oct 2 '12 at 0:43
    
It really is the additive identity. I'll edit my question. –  Rick Oct 2 '12 at 0:45
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$f\equiv 0$, maybe? –  Pedro Tamaroff Oct 2 '12 at 0:50
    
Despite the edit, the title asks for the inverse; the body, the identity. Please bring them into agreement. –  Gerry Myerson Oct 2 '12 at 0:55
    
You've already done it --- the additive inverse of the function $u$ is described/represented by $-u$. –  Gerry Myerson Oct 2 '12 at 0:59
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1 Answer

up vote 1 down vote accepted

I'm not sure if this is what you're after, but the additive identity is always given by $0\cdot\mathbf{v}$ for any $\mathbf{v}$ in the vector space.

Proof: $$\mathbf{v} = 1\cdot\mathbf{v}=(1+0)\mathbf{v} = 1\cdot\mathbf{v} + 0\cdot\mathbf{v} = \mathbf{v} + 0\cdot\mathbf{v}$$ Applying $(\mathbf{-v})$ to both sides then yields $\mathbf{0} = 0\cdot\mathbf{v}$

If the additive inverse is what you want instead, then a similar result will show that $(-1)\mathbf{v}$ is the additive inverse for $\mathbf{v}$

Proof: $$\mathbf{v} + (-1)\mathbf{v} = 1\cdot\mathbf{v} + (-1)\mathbf{v} = (1+(-1))\mathbf{v} = 0\cdot\mathbf{v} = \mathbf{0}$$ Again applying $(\mathbf{-v})$ yields $(\mathbf{-v}) = (-1)\mathbf{v}$

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It is the additive inverse that I am being asked to find. I'm stuck trying to find it for a set of continuous functions. How can I describe it for the entire set? –  Rick Oct 2 '12 at 1:08
    
I'm really not clear on what exactly you're asking here.... What would the inverse for a set in $\mathbb{R}^4$ be then? Give us an example to work with. –  EuYu Oct 2 '12 at 1:10
    
I posted an example of the solution to the vector space $P_3$ above. –  Rick Oct 2 '12 at 1:16
    
I think just writing that the additive inverse of $f$ is $-f$ is sufficient. –  EuYu Oct 2 '12 at 1:17
    
Makes sense. Thanks for your help. –  Rick Oct 2 '12 at 1:19
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