Name: Student ID: Section: 2Prof. Alan J. Laub Apr. 25, 2008Math 33A – MIDTERM EXAMINATION ISpring 2008Instructions:(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.(c) Do all 4 problems. Each problem is worth 25 points. Some partial credit may beassigned if warranted.(d) Label clearly the problem number and the material you wish to be graded.1. You are given the matrixA =1 2 0 4 10 1 0 2 10 0 0 0 2∈ IR3×5.(a) Put A in row-reduced echelon form (rref).(b) Find a basis for Im(A) from rref(A).(c) Find a basis for Ker(A) from rref(A).(d) What is rank(A)? What is rank(AT)?2. Let S = Sp111. Find a basis for S⊥.3. The vector·−47¸has components 5 and −3 with resp ect to the basis½·12¸,·31¸¾.What are the components of the vector·−47¸with respect to the basis½·01¸,·1−1¸¾of IR2?4. (a) Suppose A ∈ IR19×488, i.e., A is the coefficient matrix of a system of 19 equationsin 48 unknowns but only 8 of the equations are independent. How many linearlyindependent solutions can be found to the homogeneous linear system Ax = 0?(b) For the same A as in (a), how many linear independent solutions can be foundto the homogeneous linear system ATy =
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