We know that C is a vector space over R Define the map T C

We know that C is a vector space over R. Define the map T : C rightarrow C. such that T(z) = z, where z is the complex conjugate of z. Is T a linear map ?

Solution

Yes T is a linear map ,

Given any complex numbers z1 = x1 +iy1, z2 = x2 +iy2 V (where x1, x2, y1, y2 R) and scalar c1,c2 R, we have

T (z ) = complex conjugate of z,

T(c1*z1+c2*z2) = T((c1*x1+c2*x2)+i(c1*y1+c2*y2)) = (c1*x1+c2*x2)i(c1*y1+c2*y2) = c1*(x1iy1)+c2*(x2iy2) = c1*T(z1)+c2*T(z2). Since z1, z2 and c1,c2 were arbitrary,

hence T is a linear map (of real vector spaces).

here we use the fact that complex conjuate of real number is real number itself.