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MathematicalPhysicist

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## Main Question or Discussion Point

1) there's given a transformation f:C^n->C^n (C is the complex field), it's known that f is linear on R (real numbers) and its rank on R equals 3 i.e, dim_R Imf=3. now is f linear on C?

2) there's a function f:C->C and its known that f is linear on R, and det_R f<0, is f linear on C?

im kind of stuck on those questions, obviously in the second question, if the determinant is different than zero then the matrix is invertible (i.e has an inverse) and so it has an f^-1, so it's isomorphism on R, but im not sure if its linear on C.

now about the first question if dim_R Imf=3<n then function isnt onto C^n and thus isnt injective, but i dont know how to deduce from that about its linearity on C.

your help is apprecited.

2) there's a function f:C->C and its known that f is linear on R, and det_R f<0, is f linear on C?

im kind of stuck on those questions, obviously in the second question, if the determinant is different than zero then the matrix is invertible (i.e has an inverse) and so it has an f^-1, so it's isomorphism on R, but im not sure if its linear on C.

now about the first question if dim_R Imf=3<n then function isnt onto C^n and thus isnt injective, but i dont know how to deduce from that about its linearity on C.

your help is apprecited.