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- Although the algebra of complex vector spaces (Dimension >= 2) makes sense to me, I’m not sure about how to imagine the geometry.

According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex vectors. Therefore R2(c) is isomorphic to the 2D complex space C2.

The elements of R2(c), i.e. of C2, can be thought of, by what appears to be a slight abuse of notation, as (W+

And the elementary algebra of complex vector spaces treats elements as linear combinations of basis vectors having complex coefficients (i.e. the basis vectors need not themselves be complex).

Does this mean that:

- Every vector in C2 has its own complex plane?

OR

- Only the

OR

- Something different from either of the above.

Advice appreciated.

Thank you in advance

The elements of R2(c), i.e. of C2, can be thought of, by what appears to be a slight abuse of notation, as (W+

**W).***i*And the elementary algebra of complex vector spaces treats elements as linear combinations of basis vectors having complex coefficients (i.e. the basis vectors need not themselves be complex).

Does this mean that:

- Every vector in C2 has its own complex plane?

OR

- Only the

**vectors in C2 are complex planes?**__basis__OR

- Something different from either of the above.

Advice appreciated.

Thank you in advance