Determine whether the given set is linearly independent; if it is not, find a set of vectors with the same span that is independent.

First, we will determine if this set is linearly independent. To do this we must use the identities given in section 4.1, shown here:

With these it is easy to see that our set is not linearly independent. To show this lets convert f₁ into it’s cosh(t)+sinh(t) form.

Now that we have shown that this set is not linearly independent, we must find a set of vectors with the same span that is independent. This is pretty easy to do. We can see the last four vectors are linearly independent. We also know they generate the same span as the original five vectors. Therefore, our new set will be:

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This is great work. What a good job.


This is terrible work. Ryan's logic is severly flawed.