Hi. Define a linear mapping F: M2-->M2 by F(X)=AX-XA for a matrix A, and find a basis for the nullspace and the vectorspace(not sure if this is the term in english). Then I want to show that dim N(F)=dim V(F)=2 for all A, A≠λI, for some real λ. F(A)=F(E)=0, so A and E belongs to the nullspace. Then I define a basis for M2, as the 2x2-matrices B=(B11, B12, B21, B22) which has a 1 at i,j and 0's elsewhere. Well, this is how Im supposed to do, but it confuses me. How should I view the basis-matrix? For example with linear independency. Lets say we define A to be the 2x2-matrix with elements (a,b,c,d) and map them with F. We get 4 matrices F(Bij) and I want to sort out which ones are linearly independent, with the condition A≠λI. How do I show L.I for matrices?