Given that \( a = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(b = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\), evaluate \((2a - \frac{1}{4}b)\).

Given that \( a = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(b = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\), evaluate \((2a - \frac{1}{4}b)\).

Answer Details

To evaluate \((2a - \frac{1}{4}b)\), we first need to perform scalar multiplication on vectors \(a\) and \(b\) and then subtract the resulting vectors: \begin{align*} 2a - \frac{1}{4}b &= 2\begin{pmatrix} 2 \\ 3 \end{pmatrix} - \frac{1}{4}\begin{pmatrix} -1 \\ 4 \end{pmatrix}\\ &= \begin{pmatrix} 4 \\ 6 \end{pmatrix} - \begin{pmatrix} -\frac{1}{4} \\ 1 \end{pmatrix}\\ &= \begin{pmatrix} \frac{17}{4} \\ 5 \end{pmatrix} \end{align*} Therefore, the answer is \(\begin{pmatrix} \frac{17}{4} \\ 5 \end{pmatrix}\).