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# M1 - Math 375 Midterm Exam I In class part C1 Prove by...

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Math 375 – Midterm Exam I In class part C1. Prove by induction that for all n = 1 , 2 , 3 , . . . n X k =1 k ( k + 1) = n ( n + 1)( n + 2) 3 . (That is, 1 · 2 + 2 · 3 + 3 · 4 + ··· + n ( n + 1) = n ( n +1)( n +2) 3 . ) C2. Let S = { v 1 , . . . , v n } . (i) Write down the correct deﬁnition of the property S is linearly independent. ” (ii) Show directly from the deﬁnition in (i) that the set S R 4 given by S = n 1 2 1 0 , 1 - 2 1 0 , 0 0 0 1 o is linearly independent. (iii) Let S be as in (ii) and consider the span of S , i.e. L ( S ). Show that the space L ( S ) consists exactly of the vectors x = x 1 x 2 x 3 x 4 which satisfy x 1 = x 3 . (iv) Extend S to a basis of R 4 . Take-home part H1. Let v 1 , v 2 , v 3 be vectors in a vector space V and deﬁne w 1 = v 1 - v 2 , w 2 = v 1 - v 2 - v 3 , w 3 = v 1 + v 2 . Prove:

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M1 - Math 375 Midterm Exam I In class part C1 Prove by...

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