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Vector Spaces and Dimensions

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Problem 1. Let V be a finite-dimensional complex vector space. Then V is also a vector space over real numbers R. Show that dimV ( over R) = 2*dimV(over complex C).
Hint: If B={v1, v2, ..., vn} is a basis of V over C, show that
B'={v1, ..., vn, iv1, ... ivn} is a basis of V over R.

Problem 2. ( extend problem1) Let L be a field and let K be a subfield of L. If V is a vector space over L, then it is also a vector space over K. Prove that
dim_K(V)=[L : K]dim_L(V)
where [L:K] = dim_K(L) is the dimension of L as a vector space over K.
(Note that it is not assumed that dim_K(L)<infinity)

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