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

(c) (10 points) We further extend the model by introducing a government into the economy. The government charges a proportional tax on wage income. That is, suppose that a household's wage income is wn, then the government takes away tun from the household. The government transfers the tax revenue to all houscholds equally in a lump sum manner. How does the proportional tax on wage infuence the consumption equality? Explain your answer.

(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 

4.  Let T : be defined by T(A) = . Provethat T is a linear map, and find the matrix representation of T withrespect to the basiswhere is the transpose of A.

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）。