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May 6th, 2019, 09:05 PM   #1
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Quotient space

If W is a subspace of a vector space V over the field (Z3) Integer modulo 3 such that Dim(V)=7 and Dim(W)=4, then the number of element in V/W is how much?


I can only deduce that number of elements in the basis of V/W will be 3. How to find the total number of elements in V/W?
shashank dwivedi is offline  
 
May 7th, 2019, 04:22 PM   #2
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Originally Posted by shashank dwivedi View Post
If W is a subspace of a vector space V over the field (Z3) Integer modulo 3 such that Dim(V)=7 and Dim(W)=4, then the number of element in V/W is how much?


I can only deduce that number of elements in the basis of V/W will be 3. How to find the total number of elements in V/W?
Wouldn't it be 27? $|\mathbb Z_3| = 3$ and if $V$ has dim 7, $|V| = 3^7$. Likewise $|W| = 3^4$ and $|V/W| = 27$. That's because each coset of $W$ has the same number of elements, namely $3^4$, so that there must be $3^3 = 27$ distinct cosets.
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