Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $2^n$?

share|improve this question
2  
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

1 Answer 1

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 $2^n$.

share|improve this answer
    
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
1  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.