What does precision refer to in scientific measurements? How close a measurement is to the true value The consistency of repeated measurements The accuracy of a single measurement The number of significant figures in a measurement

Suppose \( T: \mathbb{R}^{4} \rightarrow \mathbb{R}^{3} \) is a linear transformation with \( T\left(\mathbf{e}_{1}\right)=\left[\begin{array}{c}1 \\ 5 \\ -15\end{array}\right], T\left(\mathbf{e}_{2}\right)=\left[\begin{array}{c}14 \\ -14 \\ 19\end{array}\right] \), \( T\left(\mathbf{e}_{3}\right)=\left[\begin{array}{c}-1 \\ 1 \\ 13\end{array}\right], T\left(\mathbf{e}_{4}\right)=\left[\begin{array}{c}19 \\ 9 \\ 10\end{array}\right] \). Find the (standard) matrix \( A \) such that \( T(\mathbf{x})=A \mathbf{x} \). NOTE: \( \mathbf{e}_{i} \) refers to the \( i^{\text {th }} \) column of the \( n \times n \) identity matrix.

Find the standard matrix for the linear transformation \( T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \) that reflects points about the origin and then contracts points horizontally by a factor of \( \frac{1}{4} \).

Provide a detailed explanation of the factors that influence product bundling strategies in marketing. How can businesses effectively implement product bundling to enhance their product/service mix and meet customer needs? Customer preferences, demand, and complementary items shape product bundling. This strategy enables businesses to provide cost savings and convenience to customers. Product bundling is a one-size-fits-all strategy and does not vary between businesses Product bundling is solely driven by the desire to increase sales, and businesses should bundle all their products together. The significance of product bundling varies with a company's size and doesn't necessarily cater to customer needs.

With a minimum of computations, use only a simple calculator and cofactor expansions to find the determinant of the matrix, \( M=\left[\begin{array}{cccc}2 & 0 & 0 & -9 \\ 2 & 6 & 9 & -1 \\ -2 & 0 & 0 & 0 \\ 7 & 6 & -2 & -5\end{array}\right] \)

50. Marginal Productivity The production function \( z \) for the United States was once estimated as \[ z=x^{0.7} y^{0.3}, \] where \( x \) stands for the amount of labor and \( y \) the amount of capital. Find the marginal productivity of labor and of capital. 51. Marginal Productivity A similar production function for Canada is \[ z=x^{0.4} y^{0.6} \text {, } \] with \( x, y \), and \( z \) as in Exercise 50 . Find the marginal productiv- ity of labor and of capital.