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midterm_S07 - Princeton University Department of...

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Princeton University Department of Mathematics MAT 202 – Linear Algebra with Applications MIDTERM – March 14, 2007 You have 1 hour and 30 minutes to complete your work. Please show all work , explain your answers and write neatly. Each lettered part of a problem corresponds to 5 points. Books, notes, calculators, computers are not permitted on this exam. The average on this exam was 69%. (1) Let A = 1 0 1 4 2 3 4 4 0 1 6 4 . (a) Find the reduced row-echelon form of A . (b) Determine all solutions of Ax = 0 . (c) Determine all solutions of Ax = b where the vector b is the sum of the first two columns of A . (2) Consider the matrix A = 1 0 2 0 1 1 2 1 5 . (a) Find a basis of the image of A . (b) Compute the coordinates of 3 - 1 5 in the basis you found in (a). (c) Find a basis for ( im A ) , the orthogonal complement of the image of A .
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2 (3) Let V be the subspace of R 4 with basis v 1 = 1 0 0 0 , v 2 = 1 1 0 1 and v 3 = 1 0 1 1
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