Name: Student ID: Section: 2Prof. Alan J. Laub May 1, 2009Math 33A – MIDTERM EXAMINATIONSpring 2009Instructions:(a) The exam is closed-book (except for one page of notes) and will last 50 minutes.(b) Notation will conform as closely as possible to the standard notation used in thelectures, not the textbook.(c) Do all 4 problems. Each problem is worth 25 points. Hand in the exam with workshown where appropriate, especially on problems 1, 2, and 4. Some partial credit maybe assigned if warranted.(d) Label clearly the problem number and the material you wish to be graded.1. Consider the vector space IRn×nover IR and let S be the subset of symmetric n ×n matrices (i.e., matrices A for which AT= A) and let K be the subset of skew-symmetric matrices (i.e., matrices A for which AT= −A).(a) Show that S ⊆ IRn×nand K ⊆ IRn×n.(b) Show that S ⊕ K = IRn×n. You will find the matrix identityA =12(A + AT) +12(A − AT)useful.2. Let S = Sp1112,1111. Find a basis for S⊥.3. Suppose the matrix A ∈ IRn×nnis known to have an LU factorization, i.e., A = LU.What then are the block LU factorizations, in terms of L and U , of the following2n × 2n block matrices?(a)·A 00 A¸(b)·A A0 A¸(c)·A 0A A¸(d)·A AA 0¸(e)·A AA 2A¸4. LetA =1 1 1 01 2 2 12 5 5 3∈ IR3×4.(a) Put A in row-reduced echelon form (rref).(b) What is rank(A)? What is rank(AT)?(c) Find a basis for N (A) from rref(A).(d) Find a basis for C(A) from rref(A).(e) Find a basis for C(AT) from rref(A).(f) The matrix E =2 −1 0−1 1 01 −3 1puts A in rref. Using E, find a basis for N