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01aut-f

# 01aut-f - MATH 51 FINAL EXAM(AUTUMN 2001 1 Compute the...

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MATH 51 FINAL EXAM (AUTUMN 2001) 1. Compute the following. (a) 1 2 1 2 1 0 1 0 0 - 1 (b) The angle between - 1 4 1 and 2 - 2 1 . (c) The area of the triangle with vertices (0 , 0 , 0), ( - 1 , 4 , 1) and (2 , - 2 , 1). 2. Let A = 1 2 1 2 1 3 2 4 7 18 11 22 . (a) For which vectors b = b 1 b 2 b 3 does the equation A x = b have a solution? Express your answer as one or more equations of the form ? b 1 +? b 2 +? b 3 =?. (b) Find a basis for the null space of A . (c) Find a basis for the column space of A . (d) What is the rank of A ? 3. (a) Let b = 1 2 3 4 5 v 1 = 1 3 0 1 2 v 2 = 3 5 2 1 4 v 3 = 1 0 4 3 4 . Express b as a linear combination of v 1 , v 2 and v 3 . (b) Assume A 1 2 3 4 = 2 0 - 1 and rref( A ) = 1 0 0 5 0 0 1 - 7 0 0 0 0 . Find all solutions of A x = 2 0 - 1 . 4. (a) Suppose v is a unit vector in R n . Show that, for any vector w R n , the vector w - ( w · v ) v is orthogonal to v . 1

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(b) Let T : R n R n be a linear transformation and let V = { x R n | T ( x ) = 5 x } . Show that V is a linear subspace of R n .
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01aut-f - MATH 51 FINAL EXAM(AUTUMN 2001 1 Compute the...

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