(Solved): 03) Use the definition of a linear transformation to prove or disprove that \( T \) is linear give ...

03) Use the definition of a linear transformation to prove or disprove that \( T \) is linear given that \[ T(\bar{x})=\left\langle x_{1}+2 x_{2}+3 x_{3}, x_{1}-x_{2}+4 x_{3}\right\rangle \quad \text { for } \quad \bar{x}=\left\langle x_{1}, x_{2}, x_{3}\right\rangle \]