Looking for an explanation to the following question:

Exercise 1.1.3. If V is an abstract vector space over C, then for each vector v and each k E C, obviously kv is well-defined. But as a result, for each vector v and each k E R, kv must be well-defined. So any complex vector space must also be a real vector space (but NOT vice versa). This gives rise to some very tricky distinctions. Given abstract vector spaces V, W over C, we say a map L : V - W is complex linear if L(kv) = kLv for all k E C and v E V. We say it is real linear if L(kv) = kLv for all k E R and v E V. Note that it is possible to be real linear but NOT complex linear. Given a bunch of vectors v1, . .., Uk in a complex vector space, we say they are R-linearly independent if Cavi = 0 for a1, . .., ak ER implies all a; = 0. We say they are C-linearly independent if Cav; = 0 for a1, . .., ak EC implies all a; = 0. Similarly, we can define R-spanning, C-spanning, R-basis, C-basis and so on.\f