Quotient space

Apr 2017
76
0
India
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?
 
Aug 2012
2,464
761
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.