A mapping, \( T: M_{2,2}(\mathbb{R}) \longrightarrow \mathcal{P}_{1} \) is defined by \( T\left(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\right)=(a+d)+(b-c) x \) 1.1 Show that \( T \) is a linear transformation.

To show that the mapping \( T: M_{2,2}(\mathbb{R}) \longrightarrow \mathcal{P}_{1} \) defined by \[ T\left(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\right)=(a+d)+(b-c) x \] is a linear transformation, we need to verify two properties of linearity: 1. **Additivity**: For any matrices \( A, B \in M_{2,2}(\mathbb{R}) \), we need to show that \( T(A + B) = T(A) + T(B) \). 2. **Homogeneity**: For any matrix \( A \in M_{2,2}(\mathbb{R}) \) and any scalar \( k \in \mathbb{R} \), we need to show that \( T(kA) = kT(A) \). ### Step 1: Additivity Let \[ A = \left[\begin{array}{ll} a_1 & b_1 \\ c_1 & d_1 \end{array}\right], \quad B = \left[\begin{array}{ll} a_2 & b_2 \\ c_2 & d_2 \end{array}\right]. \] Then, we have: \[ A + B = \left[\begin{array}{ll} a_1 + a_2 & b_1 + b_2 \\ c_1 + c_2 & d_1 + d_2 \end{array}\right]. \] Now, we compute \( T(A + B) \): \[ T(A + B) = T\left(\left[\begin{array}{ll} a_1 + a_2 & b_1 + b_2 \\ c_1 + c_2 & d_1 + d_2 \end{array}\right]\right) = (a_1 + a_2 + d_1 + d_2) + ((b_1 + b_2) - (c_1 + c_2)) x. \] This simplifies to: \[ T(A + B) = (a_1 + d_1) + (b_1 - c_1)x + (a_2 + d_2) + (b_2 - c_2)x = T(A) + T(B). \] ### Step 2: Homogeneity Let \( k \in \mathbb{R} \) and consider the matrix \( kA \): \[ kA = \left[\begin{array}{ll} ka_1 & kb_1 \\ kc_1 & kd_1 \end{array}\right]. \] Now, we compute \( T(kA) \): \[ T(kA) = T\left(\left[\begin{array}{ll} ka_1 & kb_1 \\ kc_1 & kd_1 \end{array}\right]\right) = (ka_1 + kd_1) + (kb_1 - kc_1)x. \] This simplifies to: \[ T(kA) = k(a_1 + d_1) + k(b_1 - c_1)x = kT(A). \] ### Conclusion Since both properties of additivity and homogeneity hold, we conclude that \( T \) is a linear transformation.