Additional Problems:

1. (This is based on the Leontief economic model.) Consider the three basic sectors of an economy: primary (Agriculature), secondary (Machinery) and tertiary (Service). We have the following information from historical data:

   1. To produce 1 unit of A, we need 0.1 unit of A, 0.2 unit of M, and 0.1 unit of S;
   2. To produce 1 unit of M, we need 0.1 unit of A, 0.3 unit of M, and 0.2 unit of S;
   3. To produce 1 unit of S, we need 0.2 unit of A, 0.1 unit of M, and 0.3 unit of S;

   Suppose the annual consumer demand from the economy is 30 units of A, 40 units of M, and 50 units of S.

   Determine the actual annual production level of A, M, and S.

2. (This is a higher dimensional version of Additional Problem #1 of Homework 1.) Consider the following transformation of vectors from \( \mathbb{R}^n \) to \( \mathbb{R}^m \) defined by

   \[ T\left(\begin{array}{c} x_1 \\ x_2 \\ . \\ . \\ x_n \end{array}\right) = \left(\begin{array}{c} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n \\ \cdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n \end{array}\right). \]

   A short hand notation of the above is given by

   \[ T\left(\begin{array}{c} x_1 \\ x_2 \\ . \\ . \\ x_n \end{array}\right) = \left(\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & & \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array}\right)\left(\begin{array}{c} x_1 \\ x_2 \\ . \\ . \\ x_n \end{array}\right). \]

1. Prove that

   \[ T\left(\begin{array}{c} x_1 \\ x_2 \\ . \\ . \\ x_n \end{array}\right) = x_1 \left(\begin{array}{c} a_{11} \\ a_{21} \\ . \\ a_{m1} \end{array}\right) + x_2 \left(\begin{array}{c} a_{12} \\ a_{22} \\ . \\ a_{m2} \end{array}\right) + \cdots x_n \left(\begin{array}{c} a_{1n} \\ a_{2n} \\ . \\ a_{mn} \end{array}\right). \]

   Note that right hand side is a linear combination of the columns of the coefficient matrix \( (a_{ij}) \).

2. Prove that \( T \) satisfies the following linearity properties (and hence \( T \) is legitimately called a linear transformation from \( \mathbb{R}^n \) to \( \mathbb{R}^m \)):

   \[ T\left(\lambda X + \mu Y\right) = \lambda T(X) + \mu T(Y) \]

   where \( X = \left(\begin{array}{c} x_1 \\ x_2 \\ . \\ . \\ x_n \end{array}\right) \) and \( Y = \left(\begin{array}{c} y_1 \\ y_2 \\ . \\ . \\ y_n \end{array}\right) \) are arbitrary vectors from \( \mathbb{R}^n \) and \( \lambda \) and \( \mu \) are arbitray numbers.