I'm trying to solve this problem. But when I search I find different situations.
My question is
V is a subspace of R³. What are the possible dimensions to V?
Thank you guys.
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Lets think about it... Can you think of a zero dimensional subspace of $\Bbb R^3$? Can you think of a one dimensional subspace? A two dimensional? Three dimensional? Four dimensional subspace of $\Bbb R^3$? Any other number larger than four? Can vector spaces have a number of dimensions that aren't a natural number or infinity according to the definitions you have been given so far (which likely involve the number of basis elements)? – JMoravitz Jun 18 '18 at 02:10
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Looks to me anywhere between 0 and 3 (inclusive).. – herb steinberg Jun 18 '18 at 02:19
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I think that now I am understanding. Thanks you!!! – OnScreen Jun 18 '18 at 02:31
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$V$ can have dimension $0,1,2$ or $3$. Take for instance the origin, for the $0$-dimensional space. You could use the spaces spanned by any $1,2$ or $3$ linearly independent vectors, as examples. Well, in the case of dimension $1$, just the span of a nonzero vector...
As to the possibility of fractional dimension, i found this.
Maybe there are more possibilities...