(a) A linear transformation is given by \[ \binom{x^{\prime}}{y_{1}}=\left(\begin{array}{ll}3 & 2 \\ 5 & 4\end{array}\right)\binom{x}{y} \] Determine (i) The image of \( (1,0) \)

To find the image of the vector \(\binom{1}{0}\) under the given linear transformation, we need to multiply the transformation matrix by the vector. The transformation is given by: \[ \binom{x^{\prime}}{y_{1}} = \left(\begin{array}{ll}3 & 2 \\ 5 & 4\end{array}\right) \binom{x}{y} \] We will substitute \(\binom{x}{y} = \binom{1}{0}\): \[ \binom{x^{\prime}}{y_{1}} = \left(\begin{array}{ll}3 & 2 \\ 5 & 4\end{array}\right) \binom{1}{0} \] Now, we perform the matrix multiplication: \[ \binom{x^{\prime}}{y_{1}} = \left(\begin{array}{ll}3 & 2 \\ 5 & 4\end{array}\right) \binom{1}{0} = \left(\begin{array}{ll}3 \cdot 1 + 2 \cdot 0 \\ 5 \cdot 1 + 4 \cdot 0\end{array}\right) = \left(\begin{array}{ll}3 \\ 5\end{array}\right) \] Thus, the image of \((1,0)\) under the linear transformation is: \[ \boxed{\left(3, 5\right)} \]