How does the Gradient Projection Method work?

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I'm trying to get this method working. I cannot understand the computations included in the answers.

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I know what $B$ and $N$ are and how they are calculated. Can't see where the basis comes from and if it's actually given. I don't even see where it's used.

Furthermore, I can't see where $(6 \ 0)$ and $(3 \ 0)$ comes from. I subbed in the starting point into the gradient matrix and got $(3 \ 6 \ 0 \ 0)$.

Then there is also this $D_N$ which I wasn't able to get to.

I tried looking up stuff online and nothing really useful was found. The notes were extremely ambiguous and messy as well.

Any help is appreciated! Thanks.