Science:Math Exam Resources/Courses/MATH221/April 2010/Question 01 (f)
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Question 01 (f) 

Let and Then the matrix U is an echelon form for A (you may assume this, you don't have to do the row reduction again.) Let be the columns of A. Express as a linear combination of the columns of 
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Hint 

Science:Math Exam Resources/Courses/MATH221/April 2010/Question 01 (f)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. 1.) To express as a linear combination of and , we take n multiples of each vector that will add up to . 2.) One way to do this is by putting the vectors in a system of equations and then putting this system of equations into an augmented matrix: + 2  2 = 4 0+ + 3 = 2 0 + 0 + = 2
3.) Reduce to row echelon form through these steps: 1.) = 3 2.) = + 2 3.) =  2 and you will get: 4.) This shows the linear combination: = 24 = 8 = 2 = 24  8 + 2 PROOF: = 24  8+ 2 =  + 