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MIT18_06SCF11_Ses1.9sol

# MIT18_06SCF11_Ses1.9sol - Exercises on independence basis...

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± Exercises on independence, basis, and dimension Problem 9.1: (3.5 #2. Introduction to Linear Algebra: Strang) Find the largest possible number of independent vectors among: 1 1 1 = , v 2 = 0 1 0 , v 3 = , 1 0 0 0 v 1 0 1 0 0 0 1 1 0 , v 5 = 1 0 and v 6 = 0 1 v 4 = . 1 1 Solution: Since v 4 = v 2 v 1 , v 5 = v 3 v 1 , and v 6 = v 3 v 2 , the vectors v 4 , v 5 , and v 6 are dependent on the vectors v 1 , v 2 and v 3 . To determine the relationship between the vectors v 1 , v 2 and v 3 we apply row reduction to the matrix [ v 1 v 2 v 3 ] : 1 1 1 1 1 1 1 1 1 1 1 1 −→ −→ 0 1 1 0 0 1 −→ 0 1 1 0 0 1 . 1

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MIT18_06SCF11_Ses1.9sol - Exercises on independence basis...

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