1. (20%) Consider the homogeneous linear system 3x- y + 2z +w =0 62- - 2y - 4z =0. Find a basis for the solution space, and find the dimension of that space.

2. (20%) Let, and Hn, be the set of all n x n symmetric matrices. Prove that Hn is a subspace of the space of n x n matrices. Find a basis for H4. What is dim(H4)?

3. (20%) Let A =.Find the inverse of A. 

4. (20%) Apply Gram-Schmidt Orthonormalization process to the vectors น1 = (1, 1,1),u2 = (1,0, -1),u3 = (2,1, -1) with respect to the stan- dard inner product.

5. (20%) . Find an orthogonal matrix Q so that QTAQ is a diagonal matrix.

1.1 Using the average instruction mix information for the program SPEC2000fp, find the percentage of all memory accesses (both data and instruction) that are for reads. Assume that two- thirds of data transfers are loads.

1.2 Compute the effective CPI for MIPS. Average the instruction frequencies for SPEC2000int and SPEC2000fp to obtain the instruction mix.

1.3 Consider an architecture that is similar to MIPS except that it supports update addressing for data transfer instructions. If we run SPEC2000int using this architecture, some percentage of the data transfer instructions will be able to make use of the new instructions, and for each instruction changed, one arithmetic instruction can be eliminated. If 25% of the data transfer instructions can be changed, which will be faster for SPEC2000int, the modified MIPS architecture or the unmodified architecture? How much faster? (Assume that both architectures have CPI values as given in the above table and that the modified architecture has its cycle time increased by 20% in order to accommodate the new instructions.)

2.1 Using C1 on both Il and 12, how much faster can the makers of I1 claim Il is compared to I2?

2.2 Which computer and compiler would you purchase if all other criteria are identical, including cost?

3.1 What will the speedup be if both improvements are made?

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