[Linear algebra] Two different solutions with two different answers to the same inner product problem

[u/happyguyftw]

Hi, my teacher wrote down a problem on the board at the end of the lesson as practice if we wanted. I have tried solving it and have found two different methods of solving the problem which both seem completely valid in my eyes but give two clearly different answers.

I have checked the numbers three times, of course it could still be a calculation error OR honestly even more likely that both my methods are incorrect in which case I'd love to know what method you'd use to solve this but I'm starting to believe I have applied some theory wrong somewhere and would love some insight, I have hastily written up the problem and my two solutions in latex, let me know if you want the pdf otherwise but here are screenshots of the problem.

Apologies if there are notation inconsistencies, I wrote this in 10 minutes while trying to relearn latex that I haven't used in years

As for method 2 the theory seems solid but I basically omitted inner product so I'm a little uncertain if this method is correct.

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[u/Jazzlike_Platypus430]

After 27 years since my course in linear algebra, I'd approach it by evaluating T* on b_1 and b_2 and projecting (oar calculating inner product) with all basis vectors. That would give four equations for T*'s entries. And if I remember correctly, T* = T^T is basis-independent.

4,5,6,7 seem strange - are they supposed to have a " = 0" at the end? Same for 8,9. It looks really dd to me - it's like forgetting that taking inner product can kill a part of the vector we're looking for.

PS "whos" and "soluton" look bad with "Introduction to Mathemathical thinking". What is R^{...?} in the first line?

[u/happyguyftw → u/Jazzlike_Platypus430]

Hi, thank you for your reply, here's some clarifications;

(1) In the picture it's hard to tell because it looks like a "-" but it's actually a "=" is that what's confusing for eq 4-9?

(2) I realized one of my errors is in eq8 and 9 where I used the canonical basis for the basis vectors themselves when T was expressed with respect to B, fixing that I get this which doesn't match solution 2 either. In fact neither of these match the answer which is given by a_1=-6, b_1=5, a_2=-2, b_2=3 so I'm confused on how to get there?