In each case, a linear transformation T : M22 M22 is defined. Give the matrix for T with

QUESTION:

In each case, a linear transformation \(T:\ M_{2\ \times\ 2} \rightarrow\ M_{2\ \times\ 2}\) is defined. Give the matrix for T with respect to the “standard basis” \(\mathbf{v}_{1}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\ \mathbf{v}_{2}=\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\ \mathbf{v}_{3}=\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right],\ \mathbf{v}_{4}=\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\) for \(M_{2\ \times\ 2}\). In each case, determine ker(T) and image (T).

a. T(X) = \(X^T\)

b. \(T(X)=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]\ X\)

c. \(T(X)=\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right]\ X\)

d. \(T(X)=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]\ X-X\ \left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]\)