## using matrices as basis

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?

Recognitions:
Gold Member
Of course, X is in the null space if and only If AX= XA. In other words, the space of all matrices that commute with A. I'm not sure what it would look like but you could "experiment" by looking at $A= \begin{bmatrix}a & b \\ c & d \end{bmatrix}$ and $X= \begin{bmatrix}s & t \\ u & v\end{bmatrix}$ and then you want to have $AX= \begin{bmatrix}as+ bu & at+ bv \\ cs+ du & ct+ dv\end{bmatrix}= \begin{bmatrix}as+ ct & bs+ dt \\ au+ cv & bu+ dv\end{bmatrix}= XA$
so that we must have as+ bu= as+ ct or just bu= ct, at+ bv= bs+ dt, cs+du= bu+ dv, and ct+dv= bu+ dv so that ct= bu. The one thing those have in common is ct= bu so that if one of t or u is 0, the other is also. If, in fact, both t and u are 0, the other equations become bv= bs so that v= s. That gives us $\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}$ as one basis vector and $\begin{bmatrix}0 & 1 \\ c/b & 0\end{bmatrix}$ as another.