Hi I'm reviewing Linear Algebra for exams and the following question came up.
Let the matrix
$\pmatrix{1+i & 3 \\ 2+i & 2-i}$
represent a linear map over the complex numbers. Find a real 4x4 matrix that represents that linear map restricting the scalars to the real numbers.
I chose the bases {(1,0),(0,1)} of C^2 over C and {1,i} of C over R, so that {(1,0),(0,1),(i,0),(0,i)} is a basis of C^2 over R. I'm not sure if what I did next is correct.
I took the vectors of the last basis and wrote their images under the map. Then I identified those basis vectors with the canonical basis of R^4. I obtained the following matrix:
$\pmatrix{1 & 3 & -1 & 0 \\ 2 & 2 & -1 & 1 \\ 1 & 0 & 1 & 3 \\ 1 & -1 & 2 & 2}$.
Is it correct?