Kunze Linear Algebra · Problem 3.2.5

Linear Transformations

Mathematics — Methods · rank-square-linear-operator-matrix

Problem

Let

C^{2 \times 2}

be the complex vector space of

2 \times 2

matrices with complex entries. Let

B = \begin{bmatrix} 1 & -1 \\ -4 & 4 \end{bmatrix}

and let

T

be the linear operator on

C^{2 \times 2}

defined by

T(A) = BA

. What is the rank of

T

? Can you describe

T^2