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Question 03 (a) 

Let be the set of all polynomials of degree at most 2. Show that is a vector space and show that form a basis of . Express the function in a vector form using B as the basis. 
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/MATH307/April 2006/Question 03 (a)/Hint 1 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. If is a vector space then the axioms must hold true
We can also think of to be in
such that commutative and associate properties are also satisfied. To show that B is a basis, we have to show that it spans and are linearly independent If we think of to be in then we can immediately identify the basis for Basis of We can use this result and build the basis of such that . Therefore
