Vector spaces like $\mathbb{R}^n$ can have different bases and we can change the basis with a matrix to get a new one. This made me wonder:
Are there any vector spaces with $dim>1$ that have only one basis?
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Spock
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3You can always add one element of a basis to another and obtain a new basis. – David Mitra Feb 06 '14 at 19:12
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I don't understand your point. – Spock Feb 06 '14 at 19:14
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3For instance if $(a,b,c)$ is a basis for $V$, then so is $(a+b,b,c)$. – David Mitra Feb 06 '14 at 19:16
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More simply, you can obtain different bases by multiplying a basis element by, say, $2.7$. – David Mitra Feb 06 '14 at 19:18
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1If we work with the field $\mathbb Z_2$ then it's a vector space over itself hence the only basis is $(\overline 1)$. – Feb 06 '14 at 19:33
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No. Let $b_1, b_2, ..., b_n$ be a basis. Create an element $a=c_1b_1 + c_2b_2+...+c_nb_n$ where not all of the $c_i$ are $0$. WLOG, we may assume $c_n \neq 0$. Then the set $b_1, b_2, ..., b_{n-1}, a$ is a basis for the space.
John Habert
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