Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.
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 .
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!
The invertibility of A is important.
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.
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 finite-dimensional 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.