Dimension of a tensor product following field extension

Question

Let E/F be a field extension (assuming finite). Let V be an n-dimensional vector space over F. How to show that the dimension of E ⊗ V over E equals n? I've set up the preliminaries, such as writing a bilinear map from E x V to E ⊗ V and choosing a basis {v_1, ..., v_n}, but don't know how to proceed.