Science:Math Exam Resources/Courses/MATH221/December 2009/Question 05 (a)
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Question 05 (a) 

Let
Find a basis for . 
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 

Science:Math Exam Resources/Courses/MATH221/December 2009/Question 05 (a)/Hint 1 
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Solution 

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We begin by forming
First thing that we need to do to matrix is find the rref: The steps needed to take to obtain the rref are as follows: 1.) Add 3 times the 1st row to the 3rd row () 2.) Add 4 times the 2nd row to the 3rd row () 3.) Add the 2nd row to the 4th row () 4.) Swap row 3 and row 4 () 5.) Add 3rd row to 2nd row () 6.) And finally add 4 times the 3rd row to the 1st row () We should obtain that:
Lastly to obtain the basis all we need to do is to interpret . The columns that are linearly independent in show us that those columns in the original equation are the vectors that form the basis for the span of . We can see that vectors: are the basis vectors. Where forms a basis for the span of 