Question: need clear solution. 3. Suppose u is a unit vector in R. . This problem is about the n by n symmetric matrix H =

need clear solution.

3. Suppose u is a unit vector in R". . This problem is about the n by n symmetric matrix H = I - 2out : (a) (5points) Show directly that H? = I. (b) (10 points) Show that H is an orthonormal matrix. (c) (10 points) One eigenvector of H is u itself. Find the corresponding eigenvalue. (d) (10 points) Is H diagonalizable? Why or why not? 4. a. A . x = b linear equation set has a non-invertable coefficient matrix A. Can we say that, linear equation set cannot be solved in this case? Explain the reason if it is unsolvable. Discuss the dimension of each possible solution space (The space has ??? dimension inside ??? dimensional space). (15 pts) b. Consider the solution space of the linear equation x+3y-5z+t=5. Does this space form a vector space? Why? How many dimension does the space have inside how many dimensional space? Explain your answer. (10 pts)

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