Let B be any non-zero fixed 2×2 matrix and T be a transformation from M 2×2 to M 2×2 defined by T(A)=BA for A€M 2×2 . If det(B)=0 then find the rank and Nullity of T.
I am getting answer as rank=1, Nullity =1 by taking B for rank and Nullity. But how to find rank and Nullity of T.
Since $B\neq0$ and $\det(B)=0$, there are linearly independent vectors $v_1=(a_{11},a_{12})$ and $v_2=(a_{21},a_{22})$ such that $B.v_1=(a,b)\neq(0,0)$ and that $B.v_2=(0,0)$. Consider the matrices$$E_1=\begin{bmatrix}a_{11}&a_{11}\\a_{12}&a_{12}\end{bmatrix},\ E_2=\begin{bmatrix}a_{11}&a_{21}\\a_{12}&a_{22}\end{bmatrix},\ E_3=\begin{bmatrix}a_{21}&a_{22}\\a_{21}&a_{22}\end{bmatrix}\text{, and }E_4=\begin{bmatrix}a_{21}&a_{11}\\a_{22}&a_{12}\end{bmatrix}.$$Then $\{E_1,E_2,E_3,E_4\}$ is a basis of $M_{2\times2}$. Compute $T(E_i)$ for each $i\in\{1,2,3,4\}$. We will see that $\dim T(M_{2\times2})=2$. Then apply the rank-nullity theorem.