Science:Math Exam Resources/Courses/MATH221/December 2011/Question 06 (b)
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Question 06 (b) 

Consider the vectors

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

We are trying to find scalars such that the following holds: Write this in matrix notation and then follow the steps from part (a). 
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Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. From Question 6 (a), we know that is orthogonal basis. Since for , it is actually orthonormal basis. Then, we have Computing the inner products, we have
and
Therefore, can be written as

Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. We create the augmented matrix and use Gaussian elimination.
We multiply the whole matrix by 2 to get
Subtracting the first row from the following three rows, we get
We rearrange the rows so that the second row becomes the last row; each of the last three rows is also multiplied by 1/2.
We subtract row 2 from row 1 and row 2 from row 3 to get
We subtract row 3 from row 1 and row 3 from row 4 to get
Multiplying row 4 by 1/2, we get
Which finally simplifies to
So our scalars will be , and . In other words . 