1. It is known based on a theorem that any set of 7 vectors in R6 will be linearly independent, does this guarantee any set of 5 vectors in R6 will be linearly independent? why?
2. Show why the set of all ordered pairs of real numbers (x, y) with addition defined as u + v = (u1 + 2v1, u2 + 2v2) and scalar multiplication as ku = (ku1, ku2) is not a vector space
2. Show why the set of all ordered pairs of real numbers (x, y) with addition defined as u + v = (u1 + 2v1, u2 + 2v2) and scalar multiplication as ku = (ku1, ku2) is not a vector space
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1...your "theorem " knowledge is false...7 in R6 is LD not LI
and a , b , a +b , c , a+b + c are 5 vectors in R6 which are not LI
#2 . ( a + b ) + c ╪ a + ( b + c )
and a , b , a +b , c , a+b + c are 5 vectors in R6 which are not LI
#2 . ( a + b ) + c ╪ a + ( b + c )