Name: Student ID: Section: 1Prof. Alan J. Laub April 29, 2011Math 33A/1 – MIDTERM EXAMINATIONSpring 2011Instructions:(a) The exam is closed-book (except for one page of notes) and will last 50 minutes. Nocalculators, cell phones, or other electronic devices are allowed at any time.(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. Some partial credit may be assigned if warranted. Labelclearly the problem number and the material you wish to be graded.(d) You must enter your discussion section above.1. (a) Consider the tridiagonal matrix A =1 1 01 2 10 1 2. Perform Gaussian elimina-tion on A to reduce it to upper triangular form U.(b) Factor A in the form LU where L is unit lower triangular.(c) What is det(A)?2. Write all solutions of the equations Ax = b where A =·1 2 31 2 4¸and b =·56¸inthe form xp+ xnwhere xpdenotes the particular solution which solves Axp= b andthe special solutions xnsolve Axn= 0.3. Consider the vector space IRn×nover IR and let U be the subset of upper triangularn × n matrices (i.e., matrices A for which aij= 0 for i > j) and let L be the subset ofstrictly lower triangular n × n matrices (i.e., matrices A for which aij= 0 for i ≤ j).(a) Show that U and L are subspaces of IRn×n.(b) Show that U ⊕ L = IRn×n.4. LetA =2 4 6 42 5 7 62 3 5 2∈ IR3×4.(a) Put A in reduced row echelon form (rref).(b) What is rank(A)? What is rank(AT)?(c) Find a basis for N (A) (the nullspace of A) from rref(A).(d) Find a basis for C(A) (the columnspace of A) from rref(A).(e) Find a basis for C(AT) from rref(A).(f) The matrix E =52−2 0−1 1 0−2 1 1puts A in rref. Using this E, find a basis forN