suppose F is a filed and G is a sub-filed of F. Suppose that as a vector space over G, F has finite dimension m. let V be a vector space over F.

suppose F is a filed and G is a sub-filed of F. Suppose that as a vector space over G, F has finite dimension m. let V be a vector space over F. show that by restricting scalar multiplication only to G, V is a vector space over G. if the dimension of V over F is n, show that the dimension of V over G is mn. Hint: There is a basis of V over F of n elements, of F over G of m elements. there is one obvious way of getting mn elements from a set of n and a set of m. see that the obvious work.