3.3_Linear_Independence_part2.pdf - Part II Example 7 Are...

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Part II
Example 7. Are the polynomials ࠵? " , ࠵? $ , ࠵? % defined by ࠵? " ࠵? = 2࠵? $ + 1, ࠵? $ ࠵? = 3 + ࠵?, ࠵? % ࠵? = −12࠵? $ + 2࠵? for all ࠵? ∈ ℝ linearly independent in % ? Set ࠵? " ࠵? " + ࠵? $ ࠵? $ + ࠵? % ࠵? % = 0 ⟺ ࠵? " ࠵? " ࠵? + ࠵? $ ࠵? $ ࠵? + ࠵? % ࠵? % ࠵? = 0 for all ࠵? ∈ ℝ ⟺ ࠵? " 2࠵? $ + 1 + ࠵? $ 3 + ࠵? + ࠵? % (−12࠵? $ + 2࠵?) = 0 for all ࠵? ∈ ℝ ⟺ (࠵? " +3࠵? $ ) + ࠵?(࠵? $ + 2࠵? % ) + ࠵? $ (2࠵? " −12࠵? % ) = 0 for all ࠵? ∈ ℝ (collect like-power terms) By the linear independence of 1, ࠵?, ࠵? $ , it must be: ࠵? " + 3࠵? $ = 0 ࠵? $ + 2࠵? % = 0 2࠵? " − 12࠵? % = 0 1 3 0 0 1 2 2 0 −12 6 0 0 0 1 0 −6 0 1 2 0 0 0 6 0 0 0 ࠵? % free There are non trivial solutions ࠵? " , ࠵? $ , ࠵? % NO , ࠵? " , ࠵? $ , ࠵? % are NOT linearly independent in % . Possible dependency relation? RREF
Example : Are the functions

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