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Unformatted text preview: MATH 20F: Final Exam Review Ari Stern March 11, 2011 Final Exam Review MATH 20F, Winter 2011, UCSD Important announcements The final exam will be held this Wednesday (March 16) from 36pm. Locations: Sections 13 (Maxx): Center 105 Sections 46 (Monty): Center 109 As before, you are allowed one sheet of letter size paper (handwritten, front and back), but no other assistance. All course and university rules on exams apply, as usual. STUDENT ID IS REQUIRED TO TAKE THE EXAM (NO EXCEPTIONS) I will be holding extended office hours on Monday, from 35 pm. Maxx and Monty may also hold extra office hours or review sessions, at dates/times to be announced. Any further announcements will be made by TritonLink email and/or the class web page. 1 Final Exam Review MATH 20F, Winter 2011, UCSD Systems of linear equations A linear system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 . . . . . . . . . . . . . . . a m 1 x 1 + a n 2 x 2 + + a mn x n = b m . This system can be represented by the augmented matrix a 11 a 12 a 1 n b 1 a 21 a 22 a 2 n b 2 . . . . . . . . . . . . . . . a m 1 a m 2 a mn b m . You should be able to: find the general solution of a system by rowreducing its augmented matrix, determine whether solutions exist (consistency), and if so, whether they are unique, know the difference between zero rows and free variables! 2 Final Exam Review MATH 20F, Winter 2011, UCSD Vector and matrix equations This system is equivalent to the matrix equation A x = b , where A = a 1 a 2 a n = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 a mn is the m n coefficient matrix , and x = x 1 x 2 . . . x n R n , b = b 1 b 2 . . . b m R m are vectors. The matrixvector product A x is defined to be the linear combination A x = a 1 x 1 + a 2 x 2 + a n x n R m of the columns of A . (This means that A x is defined only when x has the same number of rows as A has columns.) 3 Final Exam Review MATH 20F, Winter 2011, UCSD Span, column space, null space, linear independence The span of a set of vectors { a 1 , . . . , a n } R m , consists of all linear combinations a 1 x 1 + + a n x n R m . Equivalently, b span { a 1 , . . . , a n } if and only if the linear system A x = b is consistent. The span of the column vectors is also called the column space Col A R m . The null space Nul A R n consists of all solutions to the homogeneous equation A x = 0 ....
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 Spring '03
 BUSS
 Math, Linear Algebra, Algebra, Math 20F

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