Is there a unique C vector space structure on R2 such that restriction of scalar multiplication
to real numbers is the standard R2.
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The vector space R2 is represented by the usual xy plane. Each vector v in R2 has two components. The word “space” asks us to think of all those vectors—the whole plane. Each vector gives the x and y coordinates of a point in the plane : v D
R2 is not a subspace of C2 over C, since it is not closed under scalar multiplication: for example, (1,1) ∈ R2, but i(1,1) = (i, i) ∈ R2.
Your example, C2, is a 2-dimensional vector space over C, and the simplest choice of a C-basis is {(1,0),(0,1)}.
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