18.06 Problem Set 5Due Thursday, 14 October 2010 at 4pm in the undergrad math office. Please notethat the problems from the textbook are out of the 4th edition: make sure to checkthat you are doing the correct problems. For MATLAB problems, please include aprintout of your code with your problem set. You can type diary(‘‘filename’’)at the beginning of your session to save a transcript, and diary off when you aredone.Each Problem worth 10 points.1. Do problem 14 from section 8.2.2. Give all solutions of the system1 2 34 5 67 8 9x =8523. Consider the matrix A from the previous excercise. What is the rank of A? Weknow that r(AB) ≤ r(A) and r(AB) ≤ r(B), thus r(AB) ≤ min{r(A), r(B)}.What can be the rank of B if r(AB) = min{r(A), r(B)}. Give examples forB for all possible ranks. If the above equality cannot hold for some rank of Bprove it.4. Do problem 11 from section 4.1.5. Do problem 17 from section 4.1.6. Call a square matrix A orthogonal if all of its columns are of length 1, and areorthogonal to each other.(a) What is the product AAT?(b) Prove that if A is orthogonal then so is AT.(c) Suppose that A and B are orthogonal matrices. Prove that AB is or-thogonal too.7. Do problem 5 from section 4.2.8. Do problem 14 from section 4.2.9. Do problem 17 from section 4.2.110. Write a program in MATLAB or your favorite language, to Project a vector bonto the column space of a matrix A with independent columns. Try perhapsA=randn(3,2). What happens numerically if b is in the nullspace of this A?What happens when you run the program on a matrix A where the columnsare not independent? What happens if one column of A is nearly dependenton the others? (Add .0001 to a linear combination of the others, for
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