Lab8_sol_f08

# Lab8_sol_f08 - Lab#8 Solutions(week of November 3 2008 1...

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Math 115 SE Lab #8 Solutions (week of November 3, 2008) 1. Find all subsets of the following set that form a basis for 3 : () ( ) ( ) {} 1 , 1 , 2 , 3 , 7 , 2 , 1 , 1 , 4 , 2 , 3 , 1 To form a basis for 3 , we require 3 linearly independent vectors in 3 . We can check for linear independence using determinants (inserting the vectors as columns). 0 3 1 2 7 1 3 2 4 1 = = K These 3 vectors are linearly dependent and do not form a basis of 3 . 30 1 1 2 1 1 3 2 4 1 = = K These 3 vectors are linearly independent and form a basis of 3 . 30 1 3 2 1 7 3 2 2 1 = = K These 3 vectors are linearly independent and form a basis of 3 . 60 1 3 1 1 7 1 2 2 4 = = K These 3 vectors are linearly independent and form a basis of 3 . Alternatively row reduce the matrix 1- 4- 22 317 1 -2 1 -3 1 to 1020 0110 0001 and view from here which groups of 3 vectors will row reduce to RREF = Identity. 2. Let 2 1 , X X be a linearly independent set. Show that the set { } 2 1 2 1 , X X X X + is also linearly independent.

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## This note was uploaded on 09/17/2011 for the course MATH 115 taught by Professor Dunbar during the Fall '07 term at Waterloo.

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Lab8_sol_f08 - Lab#8 Solutions(week of November 3 2008 1...

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