Let $X = \lbrace x_1, x_2, . . .,x_n \rbrace$. Find a basis for the space $\mbox{Map}(X,\mathbb R)$.

Note: $Map(X,R)$ is the space of all functions that goes from $X$ to $\mathbb R$.

Note 2: I was thinking, one basis for that space can be $(1,0,...,0), (0,1,...,0) ,...,(0,...,1)$?

  • $\begingroup$ I don't understand, you are trying to find a basis for which vector space? $\endgroup$ – la flaca Apr 10 '16 at 18:23
  • $\begingroup$ Yes, i trying to find a basis for the space. $\endgroup$ – LilianaX ArangurenX Apr 10 '16 at 18:24
  • $\begingroup$ For which one? The space of all functions from $X$ to $\mathbb{R}$? $\endgroup$ – la flaca Apr 10 '16 at 18:31
  • $\begingroup$ yes...! thanks for you interest $\endgroup$ – LilianaX ArangurenX Apr 10 '16 at 18:32

More precisely, it is made up of the functions $f_i$ defined by \begin{align*} f_i\colon X &\longrightarrow \mathbf R\\ x_i&\longmapsto 1\\ x_j&\longmapsto 0\quad(j\ne i). \end{align*}

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  • $\begingroup$ Yes, thanks for clear out $\endgroup$ – LilianaX ArangurenX Apr 10 '16 at 18:23
  • $\begingroup$ sorry, then that would be a basis for space (1,0,...,0) + (0,1,...,0) + (0,...,1) ?? $\endgroup$ – LilianaX ArangurenX Apr 10 '16 at 18:31
  • $\begingroup$ @LilianaXArangurenX $(1,0,...,0)+...+(0,...,1)$ is not a space, it is just a vector (the sum of all those vectors). $\lbrace (1,\ldots ,0), \ldots, (0, \ldots ,1)\rbrace$ is a basis of $\mathbb{R}^m$ but you are asking for a basis of the space of all functions from a finite set $X$ to the reals, so the elements of your basis must be functions, the one given by Bernard is the canonical one. You can check that it generates any functions of your space and it is LI. $\endgroup$ – la flaca Apr 10 '16 at 18:49
  • $\begingroup$ Writing $(1,0,\dots,0)$ and the like is just a way to denote these functions with the list of their values, in the same order as the elements of $X$. $\endgroup$ – Bernard Apr 10 '16 at 19:34
  • $\begingroup$ Thanks Eliana and Bernard! $\endgroup$ – LilianaX ArangurenX Apr 10 '16 at 19:46

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