Science:Math Exam Resources/Courses/MATH221/December 2007/Question 06 (c)
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Question 06 (c) 

Suppose the standard matrix of a linear transformation is
Find the matrix of T with respect to the basis , i.e. find . 
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 2007/Question 06 (c)/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. Let denote the standard basis of , where and and denote the given basis by , where and Let us briefly recall how given (the matrix of T with respect to ), we can find (the matrix of T with respect to ). We first find the linear relation between the basis and :
Letting the above equality gives us the following relation for the coordinates of a vector with respect to the two bases:
Now, by definition, we have and . Therefore, we can write the following string of equalities:
Multiplying by in the above gives us . As this equality is true for every v, we conclude that
For this question, we are given and we can compute Putting these together, we find that
