Science:Math Exam Resources/Courses/MATH221/December 2008/Question 10 (a)
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Question 10 (a) 

True or False (give a proof if true, or provide a counterexample if false): Suppose that are a basis for , and A is an invertible n x n matrix. Then the vectors are also a basis for . 
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Hint 

The invertibility of A is important. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The statement is true. To prove that is a basis for , we actually only need to verify that are linearly independent vectors. (Why? Recall that for a finitedimensional vector space V, a linearly independent list of vectors with length equal to the dimension of V is a spanning list, and hence a basis. In our case and dimV = n.) To prove linear independence, assume
where . We want to show that . To begin, write the above equation as
We are given that is invertible, so apply to the above equation to find
Now the vectors are linearly independent because they are a basis. This means that the only solution to is , which is exactly what we wanted to show. Therefore is a linearly independent list of vectors with length , and thus a basis. 