please tell me how $\mathbb{R}^n$ has finite basis ? I understand the polynomial part but not the $\mathbb{R}^n$.
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1$\begingroup$ It means a basis has finitely many elements. $\endgroup$– Adam HughesCommented Nov 18, 2014 at 19:18
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$\begingroup$ i do understand that , but how Rn has finite basis ?? $\endgroup$– Mohamed OsamaCommented Nov 18, 2014 at 19:18
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$\begingroup$ @MohamedOsama: Do you agree that $\mathbb R^2$ has a basis with two elements, for example? Such as $\{(1,0),(0,1)\}$. $\endgroup$– hmakholm left over MonicaCommented Nov 18, 2014 at 19:20
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$\begingroup$ Ever consider trying to find a basis for Rn that has more than n vectors? Give it a try. $\endgroup$– JB KingCommented Nov 18, 2014 at 19:20
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$\begingroup$ yes i do it has dimension = 2 $\endgroup$– Mohamed OsamaCommented Nov 18, 2014 at 19:21
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1 Answer
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What it says is that for every $n\ge 0$ the space $\mathbb R^n$ has a finite basis.
There is not a single space called "$\mathbb R^n$" in and of itself -- there is only $\mathbb R$, $\mathbb R^2$, $\mathbb R^3$ and so forth. Speaking about $\mathbb R^n$ is just a way to make the same claim about each of these spaces separately.
answered Nov 18, 2014 at 19:24
hmakholm left over Monicahmakholm left over Monica
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$\begingroup$ why we do not use the same logic for the polynomials as you did for the Rn $\endgroup$ Commented Nov 18, 2014 at 19:25
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2$\begingroup$ @MohamedOsama: There is one space of all polynomials. That is a completely different situation from $\mathbb R$, $\mathbb R^2$, $\mathbb R^3$ and so forth being different vector spaces. $\endgroup$ Commented Nov 18, 2014 at 19:27
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$\begingroup$ so " polynomials " are only like one space but there is no finite basis will contain all of them but in the other hand Rn for every n is in itself a space , although i wonder why discrimination :D why we do not say polynomials of degree 2 are space in themselves. $\endgroup$ Commented Nov 18, 2014 at 19:30
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2$\begingroup$ @MohamedOsama: He's using standard terminology. Saying "$\mathbb R^n$ is finite dimensional" is neither more nor less than saying "Each of $\mathbb R$, $\mathbb R^2$, $\mathbb R^3$ and so forth is finite dimensional". $\endgroup$ Commented Nov 18, 2014 at 19:32
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1$\begingroup$ On the other hand, saying something about $P$ is saying something about one particular vector space (and only that), namely the one that consists of all polynomials. $\endgroup$ Commented Nov 18, 2014 at 19:34