Let X be a fixed n times n matrix Determine whether the foll

Let X be a fixed n times n matrix. Determine whether the following function is a linear transformation: T: M_n, n rightarrow M_n, n, T(A) = AX - XA

Solution

We know that a linear transformation is a mapping T :V W between two modules (including vector spaces) that preserves the operations of addition and scalar multiplication. T(x+y) = T(x)+T(y) and T(cx) = c T(x).

Let A and B be two arbitrary vectors in M_n,n and let c be an arbitrary scalar. Then T(A+B) = (A+B)X-X(A+B)= AX +BX –XA-XB ( as matrix multiplication is distributive over addition). Also, T(A) +T(B) = AX-XA +BX-XB = AX+BX – XA-XB = T(A+B). Thus, T preserves vector addition. Further T(cA) = cAX-XcA = c(AX-XA) = cT(A). Thus, T preserves scalar multiplication also. Hence T is a linear transformation.