Trace of a Linear Operator

The trace of a linear operator is defined as the sum of the diagonal elements of the matrix representation of the operator. In this case, we are given a linear operator T on the vector space V of 2x2 matrices over the real numbers.

Matrix Representation of T

To find the matrix representation of T, we need to determine how T acts on the basis vectors of V. The standard basis for V consists of the matrices E11, E12, E21, and E22, where Eij is the matrix with a 1 in the (i, j) entry and 0s elsewhere.

Let's calculate T(E11):

T(E11) = ME11 = [1 2; 3 4][1 0; 0 0] = [1 0; 3 0]

Similarly, we can calculate T(E12), T(E21), and T(E22):

T(E12) = ME12 = [1 2; 3 4][0 1; 0 0] = [2 0; 4 0]
T(E21) = ME21 = [1 2; 3 4][0 0; 1 0] = [0 1; 0 3]
T(E22) = ME22 = [1 2; 3 4][0 0; 0 1] = [0 2; 0 4]

Matrix Representation of T

Now we can write the matrix representation of T as follows:

[T] = [[1 0 2 0]; [3 0 4 0]; [0 1 0 2]; [0 3 0 4]]

The diagonal elements of this matrix are 1, 0, 0, and 4. Therefore, the trace of T is the sum of these elements, which is 1 + 0 + 0 + 4 = 5.

Conclusion

In conclusion, the trace of the linear operator T defined by T(A) = MA, where M = [1 2; 3 4], is 5. The trace is calculated as the sum of the diagonal elements of the matrix representation of T. By finding the matrix representation of T and identifying the diagonal elements, we determined that the trace is 5.