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Let $V$ be a finite dimensional vector space over $\mathbb{R}$ of dimension $n$.

The vector space $\underbrace{V \oplus V \oplus \cdots \oplus V}_k$ has dimension $kn$, and the vector space $\underbrace{V \otimes V \otimes \cdots \otimes V}_k$ has dimension $n^k$.

Is there a similar construction (that does not depend on a basis of $V$), that gives a vector space of dimension $2^n$?

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Is there any particular reason (apart from curiousity) why you ask this question? (as far as I know the answer to your question is no) – Fabian Mar 25 '11 at 18:26
Similar construction to what? The one for $k=2$ where your vectors are $(\vec{a},\vec{b})$, addition is componentwise and scalar multiplication is $c(\vec{a},\vec{b})=(c\vec{a},c\vec{b})$? – Ross Millikan Mar 25 '11 at 18:35
up vote 9 down vote accepted

You might be looking for the exterior algebra of $V$. Its definition does not depend on choosing a basis, and its dimension is $2^n$.

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Which is a special case of the Clifford algebra . – Myself Mar 25 '11 at 18:34
This is exactly what I am looking for. I can't believe I forgot about it in the first place. – NymSudo Mar 25 '11 at 18:46
@Jonas: Doesn't the exterior algebra of an $n$-dimensional space have dimension $\binom{n}{2}$? – Arturo Magidin Mar 25 '11 at 18:59
the full algebra has dimension $\sum{n\choose k}=2^n$ – yoyo Mar 25 '11 at 19:01
@Arturo: What yoyo said; $\binom{n}{2}$ is the dimension of the exterior square. – Jonas Meyer Mar 25 '11 at 19:15

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