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081st_tut1sol

081st_tut1sol - MATH1111/2008-09/Tutorial I Solution 1...

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Unformatted text preview: MATH1111/2008-09/Tutorial I Solution 1 Tutorial I Suggested Solution 1. Let a 1 =   1- 2- 5   , a 2 =   2 5 6   , b =   7 4- 3   . Determine whether b is a linear combination of a 1 and a 2 . Give it a geometrical interpretation. Ans . Want to find scalars α and β so that α a 1 + β a 2 = b . In other words, α   1- 2- 5   + β   2 5 6   =   7 4- 3   . It can be expressed as a system of linear equation    α + 2 β = 7- 2 α + 5 β = 4- 5 α + 6 β =- 3 Its augmented matrix is   1 2 7- 2 5 4- 5 6- 3   which reduces via elementary row operations to   1 0 3 0 1 2 0 0   i.e. α = 3 and β = 2, or b = 3 a 1 + 2 a 2 is a linear combination of a i , i = 1 , 2. View a i as vectors (arrows) in R 3 , the sum of them after a suitable scaling yields the vector b . MATH1111/2008-09/Tutorial I Solution 2 2. Let A be a nonsingular matrix. Show that A T is also nonsingular and find the inverse of A T ....
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081st_tut1sol - MATH1111/2008-09/Tutorial I Solution 1...

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