**Unformatted text preview: **(0 , , 0) , (1 , 1 , 1) , (1 , 2 , 3) } . 6. Prove that if ~x,~u,~v ∈ R n and a, b ∈ R are scalars, then a) ( a + b ) ~x = a~x + b~x . b) ~x · ( a~v + b~u ) = a ( ~x · ~v ) + b ( ~x · ~u ). 7. Consider the statement: “If ~a · ~ b = ~a · ~ c , then ~ b = ~ c .” a) Determine, with justiﬁcation, if the statement if true or false. b) If we specify ~a 6 = ~ 0, does that change the result? 8. Let ~x,~ y ∈ R n . Prove that k ~x + ~ y k 2 = k ~x k 2 + k ~ y k 2 if and only if ~x · ~ y = 0. 1...

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