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I know the dimension of a straight line, a circle is 1. But how can we prove it by using vector space? The dimension a vector space is the number of elements in the basis.

YuiTo Cheng
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Prince Khan
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    To more fully answer that, you're gonna need a concept of "dimension" that's different from the one you know in linear algebra. – AJY Nov 16 '16 at 05:51
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    How ? Can you explain this ? – Prince Khan Nov 16 '16 at 05:59
  • In linear algebra, we talk about vector spaces, and the dimension of a vector space. A vector space is defined by the ability to add things, and a scalar field (typically the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$) by which we can multiply. Now think about a circle in the plane. Your conception of a circle is not "vector spacey", in that you aren't thinking about adding things. If anything, we'd think a circle two-dimensional, because we wouldn't even think of circles in $1$-space (the line). (cont.) – AJY Nov 16 '16 at 06:09
  • So we instead want a different way to talk about the dimension of objects other than vector spaces (e.g. a circle) that still conform with our intuition of what the dimension should be. One way to do this is hinted at in MattG88's answer, where we think of the circle as a very nice function from $1$-space into $2$-space (this involves what are called manifolds). Another is called Hausdorff dimension, which is how we talk about the dimension of "weird" sets, like fractals, or assign to dimension to the set of all rational numbers, or of all irrational numbers. – AJY Nov 16 '16 at 06:16
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    Actually a curve which is not a stright can be thought like a stright line, but later is the topological equivalence.... I just wanted to knew that, is there any basis of such non-stright curves like stright line has a basis consisting on a singel non zero element. – Prince Khan Nov 16 '16 at 06:19
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    How can I explain the concept of dimension to a students of software engeenring? – Prince Khan Nov 16 '16 at 06:21
  • Two responses: (i) topology is not a vector space concept (there are "standard" ways to topologize certain vector spaces, but I can topologize $\mathbb{R}^2$ where the circle and line look nothing alike, or where almost anything looks the same as almost anything else), while the dimension under discussion here is; and (ii) not really because circles aren't additive structures, i.e. we don't define circles linear-algebraically, so there's not really a way to connect a linear-algebraic dimension with the intuitive dimension of a circle. – AJY Nov 16 '16 at 06:28
  • I think any connection you wanna make between linear algebra dimension and more proper geometric dimension will have to involve a linear-algebraic concept of a line, and then a geometric connection between circles and lines. – AJY Nov 16 '16 at 06:30
  • @PrinceKhan Maybe you can think like this: if you have a vector space described by $(x,0,0)$ you can say that is one dimensional (a base is (1,0,0)), it depends by a parameter ($x$) so if you have a geometric object that is described by a parameter it is one dimensional...it is a sort of analogy – MattG88 Nov 16 '16 at 06:45
  • @PrinceKhan However I agree with AJY so pay attention to the concept of dimension – MattG88 Nov 16 '16 at 06:46

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The line is 1 dimensional because you can take an element (a number $n\ne 0$) as base and you can obtain all the others number by multiplication; in this case the base is made by only one element.

You can describe a circle with radius $r$ by $re^{i\theta}$, so it depends by only a parameter ($\theta$; r is fixed), that it is just a number so the circle is like the line, it is one-dimensional.

MattG88
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    thats fine... Now we know that a curve in the plane (any curve of a continuos function in one variable) is also one dimensional... what will be the basis in this case ? – Prince Khan Nov 16 '16 at 05:54
  • As the circle the curve can be described by a parameter so it is one dimensional...we can say that the curve is like the real line – MattG88 Nov 16 '16 at 06:00
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    Yes that was my confusion.... Actually a curve whichis not a stright can be thought like a stright line, butlater is the topological concept.... I just wanted to knew that, is there any basis of such non-stright curves like stright line has a basis consisting on a singel non zero element... – Prince Khan Nov 16 '16 at 06:17
  • Oh yes we need topology...with topology we can build bridges between objects that they seem different at the first sight – MattG88 Nov 16 '16 at 06:26