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.

(a) (5 points) Solve for the equilibrium consumption, c, working hours, n, and wage rate, w.

(b) (5 points) Now we extend the benchmark model to assume that housebolds are heterogeneous. Consider that half of the bouseholds are type 1, and half of the households are type 2, Type 1 and type 2 value consumptions differently, and this is captured by their difference in the parameter 7. Type I households' utility is and type 2 households' utility is

(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.

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. 

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？