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Im new to linear algebra, so please just dont blast me.

If i have two linear spaces, with different names and equal dimensions. The two vector spaces are identical, apart from the name ?

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Yes, in the sense that if your spaces are $V$ and $W$, then there is a linear bijection from $V$ ont $W$, whose inverse is, of course, also a linear bijection. So, basically, yes, they are the same thing. More formally: even if $V\neq W$, $V$ and $W$ are isomorphic.

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  • $\begingroup$ @Poiera: Maybe an example helps: Looking at these two 3-dimensional real vector spaces: 1) R^3 with basis e.g. (1, 0, 0), (0, 1, 0), (0, 0, 1) and 2) polynomials with real coefficients and degree less than 3 with basis e.g. 1, x, x^2. As José Carlos Santos pointed out: They are isomorphic as algebraic structures - but different in their "realisation". $\endgroup$ – Bernd Jan 7 '18 at 14:47

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