4.  Let T : be defined by T(A) = . Prove
that T is a linear map, and find the matrix representation of T with
respect to the basis

where is the transpose of A.

(d) (5 points) Now we go back to the benchmark economy with homogenous households; that is, now the parameter 7 is the same among households, and there is no government, tax, or transfer. However, we consider that firms are heterogeneous. That is, there are two types of firms. Type 1 firms' production function is  and A1 ＞ A2. Solve for the equilibrium wage rate and type 1 firms' and type 2 firrns' labor hiring in units of working hours.

1. Apply the Gram-Schmidt process to transform the vectors (1,1,1), (0,1,1),(1,0, 1) into orthonormal vectors with respect to the standard inner product.

2.  . Find an invertible matrix S that diagonalizes A, and compute AS. Apply the result, compute , where is defined by 

3.  Find a basis for the solution space of the following linear system, and find the dimension of the solution space. 

5. State the Cauchy-Schwarz inequality in . Applying the inequal- ity to show 9 ≤ (a+b+c) for all positive real numbers a, b, c, and also x + 2y + 3z ≤ √14 on the sphere x2 + y2 + z2 = 1.

1.真菌（fungi）作為微生物有何特徵？真菌對人類、對自然有何好與不好的影響？ 

2.海洋中大多數的細菌（bacteria）都不能夠被人工培養，請問原因何在？怎樣提 升可培養細菌的比例？ 

3.請說明次世代定序？怎樣應用次世代定序於臨床病原菌的分析上？ 

4.微生物異常多樣化，請挑選一種（one species）微生物，怎樣使用這種微生物來 產生商業利益？請評估自身的優勢（Strengths） 、劣勢（Weaknesses） 、外部競爭 上的機會（Opportunities）和威脅（Threats）。

1. 設 X 為隨機變數其 pdf f(x) 和 cdf F(x). For a fixed number x0, define the function . Pove that g(x) is a pdf.