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Instructions and problems for Math 2210 Prelim 1, taken on September 29, 2015. The exam covers linear equations, matrices, and vector spaces. nine discussion sections, instructions for the exam, and four problems with subparts. Students are expected to show their work and not use calculators or cell phones during the exam. Academic integrity is expected.
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Discussion 1 (12:20-1:10, Frederik De Keersmaeker) Discussion 2 (1:25-2:15, Benjamin Hoffman) Discussion 3 (1:25-2:15, Frederik De Keersmaeker) Discussion 4 (1:25-2:15, Ian Lizarraga) Discussion 5 (2:30-3:20, Benjamin Hoffman) Discussion 6 (2:30-3:20, Frederik De Keersmaeker) Discussion 7 (3:35-4:25, Benjamin Hoffman) Discussion 8 (3:35-4:25, Ian Lizarraga) Discussion 9 (7:30-8:20, Ian Lizarraga)
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, v 2 =
, v 3 =
, and v 4 =
(^) in R^3.
(a) Solve the system of linear equations Ax = b for A the 3 × 4 matrix [v 1 v 2 v 3 v 4 ] and b =
(b) Interpret part (a) in terms of writing b as a linear combination. (c) Using the answer to part (a), write down a solution to Ay = 0 without re-doing the row reduction process. (d) The list of vectors v 1 , v 2 , v 3 , v 4 is linearly dependent. For each of the four vectors, if we remove it from the list, will the remaining three vectors be linearly independent? Justify your answer in each case.
0 − 1 e 0 0 f
(a) Write down a statement of the form “A is invertible if and only if...” giving a condition on the parameters e and f. (b) In the cases where A is invertible, compute its inverse. (c) Suppose that e = −1 and f = 0; your statement in (a) should tell you A is not invertible. This means the linear transformation T (x) = Ax is not one-to-one, and is not onto. Provide examples of vectors that demonstrate this is true: find x 6 = y with T (x) = T (y), and find z that’s not the image of anything by T.
If A =
(^) and AB =
(^) determine the matrix B.
This is an extra page for the solution of problem 4, if needed.
(a) Define what is LU factorization of a matrix. Compute the LU factorization of the matrix A =
(b) Give example of an invertible matrix which does not have a LU factorization. Explain carefully why such factorization does not exist.
This extra blank page for scratch work is on purpose.