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practice_exam2

practice_exam2 - MATH 3000 EXAM 2 PRACTICE SHEET...

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MATH 3000 EXAM 2 PRACTICE SHEET Computation (1) Find a basis for Span (1 , 0 , 1 , 2) , (2 , 3 , 0 , 0) , (1 , 3 , - 1 , - 2) (2) Find a basis for Span (1 , 0 , 1 , 2) , (2 , 3 , 0 , 0) , (1 , 3 , - 1 , - 2) (3) Consider the following system of equations: x 1 + x 3 + 2 x 4 = 0 2 x 1 + 3 x 2 = 0 x 1 + 3 x 2 - x 3 - 2 x 4 = 0 (a) Solve (write down the general solution to) the system of equations above. (b) Let V = ~x = ( x 1 , x 2 , x 3 , x 4 ) ~x is a solution to the system above . Find a basis for V . (4) Find conditions on the coordinates of the vector b = ( b 1 , b 2 , b 3 , b 4 ) which describe when the following system is consistent: 1 2 1 0 3 3 1 0 - 1 2 0 - 2 x = b (5) For which values of a and b does the vector (2 , 0 , a, b ) lie in Span (1 , 0 , 1 , 2) , (2 , 3 , 0 , 0) , (1 , 3 , - 1 , - 2) (6) Determine if the following collections of vectors are independent or dependent: (a) (1 , 0 , 2 , 3) , (2 , 1 , 1 , 4) , (0 , 1 , 2 , 1) , (1 , 0 , - 3 , 0) (b) (2 , 1 , 2) , (1 , 1 , 1) , (3 , 1 , 1) (c) (2 , 1 , 2 , 3 , 2) , (1 , 1 , 1 , 3 , 2) Proofs Useful facts for some of these: dim( R ( A )) = dim( C ( A )) = rank ( A ) #columns of A = rank ( A ) + nul ( A ) where the nullity of A , written nul ( A ) is the dimension of N ( A ) (this is also the number of free variables in the reduced echolon form — the rank is the number of pivot variables).
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